SOLID    GEOMETRY 


DEVELOPED   BY  THE 


SYLLABUS  METHOD 


BY 
EUGENE    RANDOLPH   SMITH,  A.M. 

HEADMASTER,  THE  PARK  SCHOOL,  BALTIMORE,  MARYLAND 

(FORMERLY  HEAD  OF  THE  DEPARTMENT  OF  MATHEMATICS,  POLYTECHNIC 

PREPARATORY  SCHOOL,  BROOKLYN,  N.T.) 


IN  CONSULTATION  WITH 

WILLIAM    H.   METZLER,   PH.D. 

DEAN    OF    THE    GRADUATE    SCHOOL,    AND    HEAD    OF    THE    DEPARTMENT 
OF    MATHEMATICS,    SYRACUSE    UNIVERSITY 


NEW  YORK  •:•  CINCINNATI  •:•  CHICAGO 

AMERICAN    BOOK    COMPANY 


COPYRIGHT,  1913,  BY 
EUGENE  RANDOLPH  SMITH. 

COPYRIGHT,  1913,  IN  GREAT  BRITAIN. 


SMITH   8YL.    SOLID   GEOM. 

W.  P.  I 


PREFACE 

THE  author  has  always  believed  that  the  teaching  of 
mathematics  should  encourage  original  thinking  on  the 
part  of  the  pupils,  and  that  a  maximum  of  thought  is 
difficult  to  obtain  when  the  pupil  is  furnished  with  a  text 
containing  the  proofs  of  the  geometrical  theorems  in 
synthetic  form.  He  has  been  using  the  heuristic  method 
for  thirteen  years,  and  has  proved  to  his  own  satisfac- 
tion that  the  average  pupil  can  enter  intelligently  into 
the  class  development  of  new  propositions,  and,  after  a 
certain  amount  of  training,  can  analyze  propositions  of 
ordinary  difficulty  with  little  or  no  help  from  the  teacher. 

This  book  has  been  written  to  encourage  what  is  often 
called  the  "  genetic,"  "  heuristic,"  or  "  syllabus  "  method  of 
teaching,  but  what  is,  in  effect,  simply  the  development  of 
new  work,  partly  in  class,  the  rest  by  assignment  as  orig- 
inal exercises  for  preparation  before  the  recitation.  It  is 
intended  to  contain  all  that  a  pupil  really  needs,  except 
that  which  the  teacher,  and  only  the  teacher,  can  best 
supply.  While  the  proofs  are  not  in  the  book  in  full  and 
formal  fashion,  the  author  has  given  as  much  analysis, 
suggestion,  and  guidance  as  he  feels  to  be  wise.  In  fact,  he 
believes  that  even  this  amount  of  help  should  often  not  be 
used  until  after  the  class  discussion  has  covered  the  topic. 

The  list  of  propositions  is  based  on  the  recommendations 
of  the  various  committees,  and  especially  on  that  "of  the 
National  Geometry  Committee  of  Fifteen.  An  attempt 

ill 


iv  PREFACE 

has  been  made  to  reduce  the  list  to  a  pedagogical  minimum, 
without  injuring  the  subject  by  over- reduction.  The 
space  thereby  saved  has  been  devoted  to  valuable  matter 
not  often  found  in  geometries,  such  as  the  preliminary 
chapter,  the  detailed  summaries,  the  Appendix,  and  the 
college  examination  questions. 

Solid  geometry  is  an  especially  promising  field  for 
heuristic  methods  because  the  pupils  studying  it  are  of 
sufficient  maturity  to  enable  them  to  think  with  some 
assurance  and  originality.  If  a  pupil  is  ever  to  obtain  the 
proper  training  in  logical  methods  of  thinking,  it  ought 
not  to  be  longer  delayed.  Besides  this,  the  results  in 
solid  geometry  in  schools  and  colleges  show  that  former 
methods  have  not  proved  entirely  satisfactory.  This  con- 
dition is  probably  due  to  the  fact  that  the  student  is  often 
plunged  into  the  subject  with  little  or  no  understanding 
of  the  figures  with  which  he  is  to  deal,  the  method  he 
should  use  in  studying  them,  or  the  relation  of  this  new 
subject  to  the  mathematics  that  has  gone  before.  This 
book  attempts  to  better  this  condition  by  introducing  a 
preliminary  chapter  that  shows  the  relation  between  plane 
geometry  and  solid  geometry^  and  by  building  up  the  solid 
geometry  figures  in  easy  gradation  from  points,  lines,  and 
planes.  This  arrangement  will  give  the  pupil  a  clear 
understanding  of  the  formation  of  the  figures  with  which 
he  is  dealing  and  will  lead  him,  in  progressive  stages,  from 
the  less  complicated  figures  and  proofs  to  those  of  greater 
difficulty.  Incidentally,  many  of  the  difficult  parts  of 
the  subject  have  been  simplified,  and  the  pupil  is  given  a 
broad  working  knowledge  of  the  properties  and  formulas 
of  solids.* 

There  are  three  combinations  of  related  parts  of  the 
subject  into  single  sections  :  prisms  and  cylinders,  pyra- 


PREFACE  V 

mids  and  cones,  and  polyhedral  angles  and  spherical  poly- 
gons. The  simplification  of  the  subject  by  these  combi- 
nations is  quite  marked,  and  will  be  evident  on  inspection 
of  the  sections  in  question.  In  the  last  case,  it  gives  op- 
portunity for  some  consideration  of  the  way  in  which  each 
can  be  made  to  depend  on  the  other. 

Some  use  has  been  made  of  two  of  the  most  powerful 
propositions  of  solid  geometry,  Cavalieri's  Theorem,  and 
the  Prismatoid  Formula,  but  only  as  alternate  methods. 
The  National  Geometry  Committee  recommends  that  these 
two  theorems  be  used  but  not  proved.  The  Prismatoid 
Formula  is  used  for  spherical  segments. 

The  use  of  limits  and  the  incommensurable  case  has 
been  left  to  the  discretion  of  the  teacher.  The  best 
present-day  thought  seems  to  favor  the  omission  of  as 
much  of  this  work  as  possible. 

The  exercises  are  of  wide  variety  in  subject  and  diffi- 
culty, and  include  a  set  of  about  two  hundred  and  seventy- 
five  college  examination  questions  chosen  from  recent 
papers  set  by  various  colleges,  the  New  York  State  Board 
of  Regents,  and  the  College  Entrance  Examination  Board. 
The  exercises  are  arranged,  some  after  the  theorems  to 
which  they  apply,  and  the  rest  in  general  sets. 

While  the  author  has  made  use  of  all  the  knowledge  of 
solid  geometry  that  his  study  and  teaching  have  given 
him,  still,  as  this  text  was  written  chiefly  from  classroom 
experiment,  it  is  impossible  to  give  credit  specifically  to 
authors  from  whom  the  germs  of  ideas  may  have  been 
received.  He  is  greatly  indebted  to  Dr.  William  H.  Metzler 
of  Syracuse  University,  for  his  painstaking  reading  of  the 
manuscript  and  for  his  valuable  criticisms.  Mr.  Howard 
F.  Hart  of  the  Montclair  High  School,  who  has  been 
trying  in  his  classes  some  of  the  ideas  that  appear  in 


vi  PREFACE 

this  book,  has  given  the  author  the  benefit  of  his  experi- 
ence with  them,  as  well  as  a  number  of  suggestions  on 
other  parts  of  the  book.  Mr.  Clarence  P.  Scoboria  of  the 
Polytechnic  Preparatory  School,  Brooklyn,  also  has  aided 
in  the  preparation  of  the  book  by  using  the  manuscript 
in  class,  and  so  making  a  practical  test  of  the  ideas 
represented.  Besides  those  mentioned,  the  author  wishes 
to  express  his  obligation  to  many  others  who  have  taken 
an  active  interest  in  the  preparation  of  the  book. 

EUGENE  RANDOLPH  SMITH. 


CONTENTS 


PRELIMINARY   CHAPTER 

How  TO  STUDY  SOLID  GEOMETRY       ...... 

I.     THE    USE   OF    PLANE    GEOMETRY   IN  SOLID  GEOM- 
ETRY   

II.     METHODS  OF  ATTACK 

III.     THE    REPRESENTATION   OF    SOLID   GEOMETRY  FIG- 
URES   

NOTES   . 


PAGES 
193 


194-202 
203-215 

216-231 
232 


BOOK   VI.  —  LINES   AND   PLANES 

I.     THE  PLANE '    .        .  233-236 

II.     RELATIVE    POSITION    OF    LINES    AND    PLANES    IN 

SPACE;   PARALLELS  AND  INTERSECTIONS       .         .  237-248 

III.  LINES  PERPENDICULAR  TO  PLANES    ....  249-253 

IV.  ANGLES  BETWEEN  PLANES 254-262 

V.     Locus  OF  POINTS 263-267 

SUMMARY  OF  PROPOSITIONS 267-272 

ORAL  AND  REVIEW  QUESTIONS          ....  272-274 

GENERAL  EXERCISES   .......  274-277 


BOOK  VII.  —  POLYHEDRONS,   CYLINDERS, 
AND   CONES 

I.     THE  PRISM  AND  THE  CYLINDER         ....  278-292 

II.     THE  PYRAMID  AND  THE  CONE 293-304 

SUMMARY  OF  PROPOSITIONS 305-307 

ORAL  AND  REVIEW  QUESTIONS 308 

GENERAL  EXERCISES 308-313 

vii 


viii  CONTENTS 

BOOK  VIII.  —  POLYHEDRAL   ANGLES   AND 
THE   SPHERE 

PAGES 

I.     DEFINITIONS;    SECANTS  AND  TANGENTS    .         .         .  314-324 

II.     SPHERICAL  POLYGONS  AND  POLYHEDRAL  ANGLES    .  325-343 

III.  AREA  ON  A  SPHERICAL  SURFACE       ....  344-349 

IV.  VOLUME  OF  A  SPHERE  AND  ITS  PARTS     .         .        .  350-357 

SUMMARY  OF  PROPOSITIONS 357-361 

ORAL  AND  REVIEW  QUESTIONS 362-364 

GENERAL  EXERCISES   .        .        .        .        .        .        .  364-366 

GENERAL 

SUMMARY  OF  THE  FORMULAS  OF  SOLID  GEOMETRY          .  367-369 

COLLEGE  EXAMINATION  QUESTIONS    .        .        .        .        .  370-392 

APPENDIX 

LOGIC 393 

CAVALIERI'S  THEOREM •        .  394 

THE  PRISMATOID -  396 

CIRCULAR  CYLINDRICAL  AND  CONICAL  SURFACES     .        .  397 
COMPARISON  BETWEEN  PLANE  GEOMETRY  AND  SPHERICAL 

GEOMETRY     .         .         ...        .        .         .        .  398 

THE  GENERAL  POLYHEDRON 400 

RADIAN  MEASURE          ...         .....  401 

SYMMETRY  403 


INDEX 


Light-faced  numbers  refer  to  pages  and  full-faced  numbers  to  articles. 

Cones  151,  158 

spherical  303 

Congruent  polyhedral  angles  265 
Congruent  spherical  polygons  264 
Conical  space  147 

Conical  surfaces  145,  321 

Construction  31 

Contact,  point  of  196,  197,  233 
Converse  of  statement  211,  318 
Convex  figures  102,  238 

Coplanar  figures  11 

Corresponding  angles  and 

sides  258 

Corresponding  sets  of  prisms  172 
Counter-clockwise  motion  260 

Cubes  128,  253 

Cuboids  127 

Cylinders  110,  114 

Cylindrical  space  105 

Cylindrical  surfaces  104,  321 

Decahedron  98 

Degree  of  formulas  311 

Descriptive  geometry  78 

Determined  surfaces  4,  9 

Diagonals  129 

Diameters  189 

Dihedral  angles  64 

Dimensions  129 
Directrix  104,  145 
Distance 

on  a  sphere  248 

point  to  a  plane  80 
Dodecahedrons    x                  98,  255 

Duplication  of  the  cube  332 

Edges 

64,  97,  99,  110,  151,  154,  312 

Elements  104,  145 

Equality  of  Tunes  287 

Equilateral  triangles  239 

Equivalence  124 

Excess,  spherical  278 

Explementary  angles  70 
ix 


Acute  angles 

70 

Addition  of  lunes 

286 

Adjacent  angles 

70 

Algebraic  analysis 

209 

Altitude                         111, 

155,  280 

Analysis 

203-209 

Analytical  geometry 

266 

Angles, 

between  arcs 

221 

dihedral 

64 

face 

100 

line  with  plane 

83 

of  spherical  polygon 

237 

plane  or  measuring 

65 

spherical 

222 

Area                                97, 

110,  151 

Attack,  methods  of 

203 

Axioms                                6 

,  18,  282 

Axis                               115, 

157,  204 

Bases                            110, 

151,  154 

Cavalieri  bodies 

137 

Cavalieri's  Theorem 

137,  319 

Center  lines 

232 

Center  sects 

232 

Center  of  a  sphere 

187 

Central  polyhedral  angles 

241 

Circles,  of  spheres 

203,  207 

Circular  cones 

158 

Circular  cylinders 

114 

Circular  cylindrical  and 

conical  surfaces 

321 

Circumscribed  figures 

polyhedrons 

201 

prisms 

116 

pyramids 

159 

sets  of  prisms 

172 

spheres 

195 

Classification 

203 

Clockwise  motion 

260 

Closed  surfaces 

105,  147 

Coincidence 

263 

Complementary  angles 

70 

INDEX 


Face  angles  100 

Faces    64,  97,  98,  99,  110,  151,  154 

Figures,  representation  of  216 

Foot,  of  line  14 

Frustums  153 

Generation 

of  a  conical  surface  145 

of  a  cylindrical  surface        104 

of  a  prismatic  surface  104 

of  a  pyramidal  surface         145 

of  a  sphere  188 

of  a  spherical  surface  279 

of  a  zone  281 

Generatrix  104,  145 

Great  circles  207 

Hexahedrons  98,  253 

Icosahedrons  98,  256 
Inscribed  figures 

polyhedrons  195 

prisms  116 

pyramids  159 

sets  of  prisms  172 

spheres  201 

Intersecting  planes  17 

Intersection  of  loci  207 

Isosceles  triangles  239 

Lateral  area  110,  151 

Lateral  faces  and  edges  110,  154 

Lateral  surfaces  110,  151 
Limit 

of  circumscribed  poly- 
hedron 300 
of  inscribed  polyhedron       282 
of  prism  116 
of  pyramid  159 
of  sets  of  prisms  175 
Loci  88 
Logic  210,  318 
Limes                             259,  285-296 

Material  spheres  227-230 

Measuring  angles 

of  dihedral  angles  65,  69 

of  spherical  angles  225 

Midsections  169 

Negative  converse  of  state- 
ment 210 


Negative  of  statement  210 

Non-coplanar  figures  11 

Normal  lines  and  planes  52 
Notation 

of  lunes  288 

of  spherical  polygons  299 

Oblique,  to  a  plane  52 

Oblique  cylinders  and  prisms  110 

Oblique  parallelepipeds  127 

Oblique  sections  106 

Obtuse  angles  70 

Obverse  of  statement  210 

Octahedrons                            98,  254 

Opposite  polyhedral  angles  257 

Opposite  spherical  polygons  258 

Parallel  figures 

lines  and  planes  14 

sections  106,  148 

summary  of  parallels        41,  43 

three  planes  39 

two  planes  21 

Parallelepipeds  125,  127 

Parts  100 

Pentahedrons  98 

Perpendiculars 

lines  and  planes  52 

two  planes  70 

Plane  angles  65 

Plane  geometry,  relation  to 

solid  geometry  193,  322 

Planes, 

construction  of  31 

defined  5 

intersecting  18,  34,  35,  38 

lines  in  -  6,  22 

lines  parallel  to  41,  43 

lines  perpendicular  to       50,  53 

parallel  21,  39,  41 

relative  positions  of          15,  33 

representation  of  10 

Polar  chords  215 

Polar  distances  215 

Polar  triangles  273 

Poles  204 

Polygons,  spherical  237 

Polyhedral  angles 

99,  240,  241,  244,  257 
Polyhedrons  97,  103,  195,  250,  323 
Prismatic  space  105 

Prismatic  surface  104 


INDEX 


XI 


Prismatoids,  defined 

formula 

proof 
Prisms,  defined 

inscribed  and  circum 

scribed 
Projections 
Pyramidal  space 
Pyramidal  surface 
Pyramids 

spherical 

Quadrant 

Quad  ran  tal  triangles 

Radian  measure 

Radius 

of  a  circular  cone 
of  a  circular  cylinder 
of  a  sphere 

Ratio  of  solids 

Reflex  angles 

Relative  positions 
of  line  and  plane 
of  line  and  sphere 
of  plane  and  sphere 
of  point  and  sphere 
of  three  planes 
of  two  lines 
of  two  planes 

Right  angles 

Right  cones 

Right  cylinders 

Right  parallelepipeds 

Right  prisms 

Right  sections 

Scalene  triangles 
Secants 
Sections 

Sectors,  spherical 
Segments,  spherical 
Similar  solids 
Slant  height 
Solid  angles 
Solid  geometry 
Solids,  defined 
Space 


181 

Sphere,  defined 

187 

182 

Spherical  excess 

278 

320 

Spherical  figures 

110, 

113 

angles                      222,  225, 

226 

cum- 

cones 

303 

172 

polygons                          237, 

258 

77 

pyramids 

305 

147 

sectors 

303 

145 

segments                        309, 

310 

151,  156, 

157 

surfaces 

187 

305 

triangles 

239 

wedges 

307 

217 

Squared  paper 

217 

342 

Straight  angles 

70 

Subtraction  of  limes 

286 

324 

Summaries              95,  185,  315, 

317 

Supplementary  angles 

70 

158 

Surfaces              3,  97,  104,  145, 

187 

der 

115 

Symmetric  figures 

325 

189 

Symmetric  polyhedral  angles 

131 

260, 

265 

70 

Symmetric  spherical  polygons 

260, 

264 

14 

Synthesis 

203 

! 

196 

re 

197 

Tangent 

re 

193 

to  a  cone 

162 

33 

to  a  cylinder 

118 

13 

to  a  sphere                     196, 

197 

15 

two  spheres 

233 

70, 

239 

Tetrahedrons                          98, 

252 

158 

Transversals 

28 

110, 

114 

Triangles,  spherical 

239 

127 

Trirectangular  triangles 

348 

110 

Truncated  figures 

152 

106 

Ungula 

307 

239 

Unit  of  volume 

135 

196, 

197 

101,  106, 

148 

Vertical  dihedral  angles 

70 

303 

Vertical  polyhedral  angles 

257 

309 

Vertices                     97,  99,  146, 

237 

312 

Volume                                    97, 

135 

157, 

158 

99 

Wedge 

312 

193,  8, 

322 

spherical 

307 

2 

1 

Zones                                     280, 

281 

SUMMARY   OF   GEOMETRICAL   SIGNS 


-f-  plus,  sign  of  addition 
—  minus,  sign  of  subtraction 
X  times,  sign  of  multiplication 
-f- ,  /,  : ,  divided  by,  sign  of  division 
•y/  square  root  sign 
=  is  (or  are)  equal,  or  equivalent 
=£  is  not  equal,  or  equivalent,  to 
=  is  identical  to 
^  is  congruent  to 
=  approaches  as  a  limit 
~  is  similar  to 
>  is  greater  than 
>  is  not    reater  than 


<  is  less  than 

«jC  is  not  less  than 

II  is  parallel  to 

_L  is  perpendicular  to 

Z.  or  2£  angle 

A  triangle 

O  parallelogram 

HH  rectangle 

Q  square 

O  circle 

^  arc 

.•.  therefore 

• .  •  because,  since 


The  signs  for  figures  become  plural  by  the  addition  of  s,  often  within, 
the  sign,  as  HJ  for  rectangles. 


xii 


SOLID  GEOMETRY 

PRELIMINARY   CHAPTER 

HOW   TO    STUDY    SOLID    GEOMETRY 

The  Relation  between  Solid  Geometry  and  Plane  Geom- 
etry. —  Solid  geometry  is  not  an  entirely  new  subject. 
It  is  simply  an  extension  of  plane  geometry,  with  which 
it  has  much  in  common.  It  is  true  that  the  conditions 
are  somewhat  changed  by  the  use  of  figures  not  entirely 
in  one  plane,  but  the  large  part  of  solid  geometry  that 
treats  of  single  planes  of  a  solid  figure  introduces  very 
little  that  is  really  new,  since  propositions  of  plane  geom- 
etry obviously  hold  for  any  one  plane  of  a  figure.  Even 
in  the  part  of  the  subject  where  planes  are  least  used,  the 
definitions,  axioms,  and  propositions  of  plane  geometry 
are  constantly  applied,  either  directly  or  indirectly.  As 
the  methods  of  attacking  a  new  proposition  in  solid  geom- 
etry are  the  same  as  those  used  in  plane  geometry,  the 
solid  geometry  propositions  are  made  to  depend,  as  far  as 
possible,  directly  on  plane  geometry  propositions. 

In  order  that  the  pupil  may  find  the  transition  to  figures 
of  three  dimensions  as  easy  as  possible,  this  preliminary 
chapter  gives  some  directions  about  methods  of  work.  It 
contains  a  discussion  of  the  plane  geometry  propositions 
that  can  be  used  in  solid  geometry,  a  review  of  methods 
of  attack,  and  a  short  description  of  a  method  for  drawing 
solid  geometry  figures.  The  section  on  drawing  figures 
can  best  be  used  as  the  subject  is  studied,  the  different 
figures  being  examined  as  the  pupil  needs  them  in  his  work, 

193 


194  PRELIMINARY   CHAPTER 

SECTION   I.     THE   USE   OF    PLANE    GEOMETRY   IN 
SOLID  GEOMETRY 

In  starting  solid  geometry,  all  of  plane  geometry  is  at 
the  disposal  of  the  student,  but  great  care  must  be  taken 
that  it  is  used  only  for  those  figures  for  which  the  plane 
geometry  proofs  will  hold.  For  example,  it  has  been 
proved  in  plane  geometry  that  Lines  parallel  to  the  same 
line  are  parallel  to  each  other ;  but  as  this  was  proved  only 
for  the  case  when  the  three  lines  were  in  one  plane,  it  can- 
not be  used  for  the  general  case  in  solid  geometry.  It 
cannot  be  used,  for  instance,  to  prove  that  the  back  edge 
of  the  ceiling  of  a  room  is  parallel  to  the  front  edge  of 
the  floor  because  they  are  both  parallel  to  the  front  edge 
of  the  ceiling. 

The  proofs  of  some  propositions  in  plane  geometry 
would  hold  even  if  the  figures  used  did  not  lie  entirely 
in  one  plane  ;  as,  for  example,  congruence  proofs.  In  a 
second  kind  of  proposition,  as  in  theorems  concerning 
triangles,  the  figures,  from  the  conditions  of  the  proposi- 
tion, necessarily  lie  in  one  plane.  Such  propositions  can  be 
used  wherever  their  figures  occur  in  solid  geometry.  In 
a  third  kind  of  proposition,  where  there  is  nothing  in  the 
conditions  to  show  whether  or  not  the  figure  is  in  one 
plane,  it  must  be  proved  to  lie  in  one  plane  before  the 
plane  geometry  proof  can  be  assumed  to  hold.  An  ex- 
ample is  the  proposition  Lines  perpendicular  to  the  same 
line  are  parallel.  By  looking  at  the  edges  meeting  at  the 
corner  of  a  room  it  will  be  seen  that  this  proposition  does 
not  hold  for  the  general  case. 

The  propositions  of  plane  geometry  that  are  of  most 
value  in  solid  geometry  will  be  summarized  under  the 
three  heads  indicated  in  the  last  paragraph. 


USE  OF  PLANE  GEOMETRY  IN  SOLID  GEOMETRY     195 

A.  SOME  PARTS  OF  PLANE  GEOMETRY  THAT  CAN  BE  USED  IN 
SOLID  GEOMETRY  WHETHER  OR  NOT  THE  ENTIRE  FIGURE 
IS  IN  ONE  PLANE 

I.    (a)  All  the  definitions  and  axioms. 

(5)  All  right  angles  are  equal ;  all  straight  angles 

are  equal. 
(V)   Complements,    supplements,    or   explements    of 

equal  angles  are  equal. 
II.    FIGURES  CONGRUENT. 

(a)  The  definition :  two  figures  are  congruent  if 
they  can  be  made  to  coincide. 

(6)  Two  triangles  are  congruent  if  they  have  three 

parts  of  one,  of  which  at  least  one  is  a  side, 
equal  to  the  corresponding  parts  of  the 
other ;  unless  those  three  parts  are  two  sides 
and  an  acute  angle  opposite  one  of  them. 

III.  SIMILAR  FIGURES. 

(a)  Two  polygons  are  similar  (1)  if  their  corre- 
sponding sides  are  proportional  and  their  cor- 
responding angles  are  equal ;  (2)  if  they  are 
similar  to  the  same  potygon. 

(5)  Two  triangles  are  similar  if  they  have  (1)  two 
angles  equal ;  (2)  two  sides  of  one  propor- 
tional to  two  sides  of  the  other,  and  the 
included  angle  equal ;  (3)  their  sides  corre- 
spondingly proportional. 

(e)  Areas  of  similar  figures  are  proportional  to  the 
squares  of  any  two  corresponding  sects. 

IV.  OTHER  COMPARISONS. 

(a)  If  two  triangles  have  two  sides  of  one  equal  to 
two  sides  of  the  other,  then  the  third  side  of 
one  is  correspondingly  greater  than,  equal  to, 
or  less  than,  the  third  side  of  the  other,  ac- 


196  PRELIMINARY   CHAPTER 

cording  as  its  opposite  angle  is  greater  than, 
equal  to,  or  less  than,  the  opposite  angle  of 
the  other,  and  conversely. 

(6)  Areas  of  triangles  having  an  angle  of  one  equal 
to  an  angle  of  the  other  are  to  each  other  as 
the  products  of  the  including  sides. 

B.  PLANE  GEOMETRY  PROPOSITIONS  THAT  CAN  BE  USED  IN  SOLID 
GEOMETRY  BECAUSE  THE  NATURE  OF  THE  FIGURE  MAKES 
IT  LIE  ENTIRELY  IN  ONE  PLANE 

See  §  9.     Figures  that  lie  entirely  in  one  plane  are  a  tri- 
angle, a  line  and  a  point  outside,  two  intersecting  lines,  two 
parallel  lines  and  therefore  a  parallelogram,  and  a  circle. 
I.    PROPOSITIONS  ABOUT  A  TRIANGLE. 

(a)  If  one  side  of  a  triangle  is  greater  than,  equal 

to,  or  less  than,  a  second  side,  the  angle  op- 
posite the  first  side  is  respectively  greater 
than,  equal  to,  or  less  than,  the  angle  op- 
posite the  second  side,  and  conversely. 

(b)  The  sum  of  two  sides  of  a  triangle  is  greater 

than  the  third  side;  their  difference  is  less 
than  that  side. 

(c)  The  perpendicular  bisectors  of  the  sides  of  a 

triangle  meet  in  a  point  (called  the  circum- 
center)  equidistant  from  the  vertices. 

(d)  The  bisectors  of  the  interior  angles  of  a  tri- 

angle meet  in  a  point  (called  the  incenter) 
equidistant  from  the  sides. 

(e)  The  medians  of  a  triangle  meet  in  a  trisection 

point  of  each.     In  an  equilateral  triangle, 
the  medians  are  also  the  altitudes. 
(/)  A  line  parallel  to  one  side  of  a  triangle  cuts 
the    other    sides    proportionally,    and    con- 
versely ;  it  is  in  the  same  ratio  to  the  side  to 


USE  OF  PLANE  GEOMETRY  IN  SOLID  GEOMETRY     197 

which  it  is  parallel  as  the  sects  it  cuts  off  on 
the  other  sides  from  their  common  vertex 
are  to  those  sides. 

($r)  In  a  right  triangle  with  the  altitude  drawn  to 
the  hypotenuse,  (1)  the  triangles  are  simi- 
lar ;  (2)  the  altitude  is  the  mean  propor- 
tional between  the  sects  of  the  hypotenuse ; 
(3)  each  leg  is  the  mean  proportional  be- 
tween the  hypotenuse  and  its  own  projection 
upon  the  hypotenuse. 

II.    PROPOSITIONS   ABOUT    PARALLELS    AND    PARAL- 
LELOGRAMS. 

(a)  Definition  :  lines  in  the  same  plane  that  never 
meet  are  parallel. 

(5)  Through  a  point  there  can  be  but  one  parallel 
to  a  given  line. 

(c)  If  two  parallels  are  cut  by  a  transversal,  two 
sets,  each  of  four  equal  angles,  are  formed, 
the  angles  of  one  set  being  supplemental  to 
the  angles  of  the  other  set,  and  conversely. 

(t?)  In  a  parallelogram  (1)  the  opposite  sides  are 
equal,  (2)  the  opposite  angles  are  equal, 
(3)  either  diagonal  divides  it  into  congru- 
ent triangles,  (4)  the  diagonals  bisect  each 
other. 

(0)  A  quadrilateral  is  a  parallelogram  if  (1)  the 
opposite  sides  are  parallel,  (2)  two  opposite 
sides  are  equal  and  parallel,  (3)  the  diago- 
nals bisect  each  other. 
(/)  Parallelograms  having  equal  bases,  and  lying 

between  the  same  parallels,  are  equivalent. 
The  sect  between  the  midpoints  of  the  legs  of 
a  trapezoid  is  one  half  the  sum  of  the  bases. 


198  PRELIMINARY   CHAPTER 

III.  PROPOSITIONS  IN  REGARD  TO   A   LINE  AND  AN 

EXTERNAL  POINT. 

(#)  There  can  be  but  one  perpendicular  from  an 
external  point  to  a  line. 

(5)  From  an  external  point  to  a  line,  (1)  the  per- 
pendicular is  the  shortest  line;  (2)  obliques 
that  make  equal  angles  with  the  perpen- 
dicular or  witli  the  given  line,  or  that  cut 
off  equal  distances  from  the  foot  of  the 
perpendicular,  are  equal,  and  conversely; 
(3)  obliques  that  make  unequal  angles  with 
the  perpendicular,  or  with  the  given  line,  or 
that  cut  off  unequal  distances  from  the  foot 
of  the  perpendicular,  are  unequal,  the  one 
that  makes  the  greater  angle  with  the  per- 
pendicular, or  that  cuts  off  the  greater 
distance,  being  longer,  and  conversely. 

IV.  PROPOSITIONS  ABOUT  CIRCLES. 

(a)  Radii,  or  diameters,  of  a  circle  are  equal. 

(5)  The  diameter  is  the  greatest  chord ;  it  bisects 
the  circle  and  its  circumference.  If  it  is 
perpendicular  to  a  chord,  it  bisects  the 
chord,  and  conversely. 

(c)  One,  and  but  one,  circle  can  be  inscribed  in, 

or  circumscribed  about  a  given  triangle. 

(d)  A  central  angle  is  measured  by  its  arc ;    an 

inscribed  angle,  by  one  half  its  arc. 

(e)  Equal  chords  subtend  equal  arcs,  and  are  equi- 

distant   from    the   center,    and   conversely. 
The  greater  chord  subtends  the  greater  arc, 
and  is  nearer  the  center,  and  conversely. 
(/)  A  tangent  is  perpendicular  to  the  radius  drawn 
to  the  point  of  contact. 


USE  OF  PLANE  GEOMETRY  IN  SOLID  GEOMETRY     199 


A  tangent  from  a  point  to  a  circle  is  the  mean 
proportional   between   the   sects   (from  the 
point  to   the  circumference)  of  any  secant 
from  the  same  point  to  the  circle. 
(Ji)  A  circle  can  be  inscribed  in,  or  circumscribed 

about,  any  regular  polygon. 

(i)  If  the  number  of  sides  of  an  inscribed  or  a  cir- 
cumscribed regular  polygon  is  increased  in- 
definitely, the  perimeter  and  the  area  of  the 
polygon  approach  the  circumference  and  area 
of  the  circle  as  limits. 
V.  PROPOSITIONS  ABOUT  FIGURES  FORMED  BY  Two 

INTERSECTING   LINES. 
(a)    Any  two  vertical  angles  are  equal. 
(6)    Bisectors  of  vertical  angles  are  in  a  straight  line. 
(c)    A  quadrilateral  is  a  parallelogram  if  its  diago- 

nals bisect  each  other. 

VI.  CONSTRUCTIONS.  The  given  figure  is  in  one  plane, 
and  the  construction  is  to  be  done  in  that  plane, 
except  where  it  can  be  equally  well  done  in  a 
different  plane,  as  in  (d)  and  (e)  below.  Where 
the  given  figure  can  be  in  more  than  one  plane,  as  in 
(a)  and  part  of  (<?),  the  construction  must  be  done 
in  one  of  those  planes.  In  every  case  there  must 
be  one  plane  in  which  the  construction  is  done. 
To  construct 

(a)    The  perpendicular  bisector  of  a  sect. 
(6)    The  bisector  of  an  angle. 

(c)  The  perpendicular  to  a  line  from  or  at  a  point. 

(d)  A  triangle,  given  its  sides. 

(e)  An  angle  equal  to  a  given  angle. 

(/)  A  line  through  a  given  point  ||  to  a  given  line. 
(</)    The  center  of  a  circle,  given  an  arc. 


200  PRELIMINARY   CHAPTER 

(Ji)  A  circle,  given  an  inscribed  angle,  and  the 
chord  it  subtends;  especially,  the  locus  of 
the  vertex  of  a  right  angle  opposite  a  given 
hypotenuse. 

(i)     A  triangle  equivalent  to  a  given  polygon. 

(,/)  A  square  equivalent  to  (1)  the  sum  of  two  or 
more  squares,  (2)  the  difference  of  two 
squares,  (3)  any  number  of  times  a  given 
square. 

(6)  A  third  or  a  fourth  proportional  to  given  sects, 

or  a  mean  proportional  between  them. 

(7)  A   regular   polygon   inscribed  in,  or   circum- 

scribed about,  a  given  circle. 
VII.    FORMULAS. 

(a)  For  the  angles  of  triangles  or  other  polygons. 

(1)  An  exterior  angle  of  a  triangle  equals 

the  sum  of  the  non-adjacent  interior 
angles.  The  sum  of  the  angles  of  a 
triangle  is  a  straight  angle. 

(2)  The  sum  of  the  interior  angles  of  a  poly- 

gon of  n  sides  is  equal  to  (n  —  2) 
straight  angles  ;  the  sum  of  its  exterior 
angles  is  equal  to  two  straight  angles. 
Each  angle  of  a  regular  polygon  of  n 

sides  equals straight  angles. 

n 

(5)  Concerning  the  sides  of  a  right  triangle.  The 
square  of  the  hypotenuse  of  a  right  triangle 
equals  the  sum  of  the  squares  on  the  legs ; 
the  square  of  either  leg  equals  the  difference 
of  the  squares  on  the  hypotenuse  and  the 
other  leg. 

(<?)    The  altitude  of  an  equilateral  triangle  of  side 


USE  OF  PLANE  GEOMETRY  IN  SOLID  GEOMETRY    201 

-  V3 
of  sides  a,  b,  c,  is 


#,  is  -  V3;   the  altitude  on  b  of   a  triangle 


-  V«(«-  aX«-&X«-<0  where  s  =  ^  +  1  +  g* 
(c?)  The  circumference  of  a  circle  is  2  vrr,  where  r 

is  the  radius,  and  TT  =  3.1416  —  . 
(e)    Area.    The  base  is  represented  by  6,  by  b1  and  £2 
when  there  are  two  bases  ;   the  altitude,  by 
h  ;  the  perimeter,  by  p  ;  the  apothem,  by  a. 

(1)  Of  a  rectangle  or  other  parallelogram, 

is  bh. 

(2)  Of  a  triangle,  is  —  ;    of  an  equilateral 


triangle    of    side  #,  is  f^J  V3  ;    of   a 
triangle,  using  the  notation  in  part  (c), 


is  Vs(s  —  a)(s  —b)  (s  —  c). 

(3)  Of  a  trapezoid  is  -(^b1  +  52). 

(4)  Of  a  regular  polygon  is  ^p  •  a. 

(5)  Of  a  circle  is  vrr2,  where  r  is  its  radius. 

C.  PLANE  GEOMETRY  PROPOSITIONS  THAT  CAN  BE  USED  IN  SOLID 
GEOMETRY  ONLY  WHEN  THE  ENTIRE  FIGURE  CAN  BE  SHOWN 
TO  LIE  IN  ONE  PLANE 

I.    CONCERNING  PARALLELS  AND  PERPENDICULARS. 

(a)  There  can  be  but  one  perpendicular  to  a  given  line 
at  a  given  point.  Note  that  in  solid  geometry 
any  number  of  planes  can  contain  the  given 
line,  and  there  can  be  one  perpendicular  to  the 
line  at  the  given  point  in  each  of  these  planes. 

(b*)  A  line  perpendicular  to  one  of  two  parallels  is 
perpendicular  to  the  other  also. 

(c)    Lines  perpendicular  to  the  same  line  are  parallel. 


202  PRELIMINARY  CHAPTER 


Lines  perpendicular  to  intersecting  lines  must 

meet  each  other. 
II.    CONCERNING  CIRCLES. 

(a)  A  line  perpendicular  to  a  radius  at  its  end  on  the 
circumference  is  tangent  to  the  circle. 

(6)  Two  circles  are  tangent  if  they  meet  at  a  point 
on  their  center  line,  or  are  tangent  to  the  same 
line  at  the  same  point  ;  they  intersect  twice  if 
they  meet  at  a  point  not  on  their  center  line. 
III.  Loci.  The  given  figure  is  in  a  plane.  If,  as  in  (a), 
(e),  (/),  and  (#),  it  can  be  in  more  than  one 
plane,  one  of  those  planes  must  be  used.  The 
locus  of  points  in  that  plane  in  each  case  is  : 

(a)  Equidistant  from  two  given  points,  the  perpen- 
dicular bisector  of  the  sect  joining  them. 

(£)  Equidistant  from  three  non-collinear  points,  the 
circumcenter  of  their  triangle. 

(<?)  Equidistant  from  two  intersecting  lines,  the  bi- 
sectors of  the  angles  between  them  ;  from  two 
parallel  lines,  the  third  parallel  midway  be- 
tween them. 

(c?)  Equidistant  from  three  lines  meeting  in  pairs,  the 
incenter  and  three  excenters  of  their  triangle. 

(e)  At  a  fixed  distance  from  a  given  point,  the  ch'- 
cumference  of  a  circle  drawn  with  the  point  as 
center,  and  the  distance  as  radius. 

(/)  At  a  fixed  distance  from  a  given  line,  two  paral- 
lels at  that  distance  from  the  line. 

(</)  Of  the  vertex  of  a  given  angle  subtended  by  a 
given  fixed  sect,  the  two  arcs  of  circles  having 
that  sect  as  a  chord,  and  the  angle  inscribed  on 
the  arc  of  that  chord  ;  in  particular,  for  a 
right  angle,  a  circle  on  the  sect  as  a  diameter. 


SECTION   II.     METHODS   OF  ATTACK 

Synthesis  and  Analysis.  The  most  usual  form  for  a 
proof  is  synthetic,  that  is,  the  proof  starts  from  the  given 
conditions,  and,  step  by  step,  builds  on  them  until  the  con- 
clusion is  reached.  The  demonstrations  of  plane  geom- 
etry, to  which  the  pupil  is  accustomed,  are  of  this  form. 

While  this  is  an  excellent  way  in  which  to  state  a  proof 
that  is  already  known,  it  is  not  adapted  to  discovering  a 
proof,  unless  the  conclusion  is  an  almost  immediate  con- 
sequence of  the  conditions. 

Usually  a  proof  must  be  discovered  by  analysis,  that  is, 
by  working  back  from  the  required  conclusion  to  the 
conditions  upon  which  the  conclusion  must  depend.  The 
different  methods  of  analyzing  a  proposition  will  be  taken 
up  in  some  detail.  The  one  which  is  most  nearly  pure  anal- 
ysis is  called  the  analysis  method,  the  others  being  named 
from  some  important  feature  of  the  process  used. 

Analysis  by  the  Classification  Method.  The  most  gener- 
ally useful  of  all  the  methods  of  attack  is  analysis  by  the 
classification  method.  It  considers  all  propositions  as  be- 
longing to  classes  according  to  their  conclusions.  For 
example,  in  using  lines,  one  class  proves  sects  equal,  an- 
other proves  them  unequal,  a  third  proves  them  propor- 
tional, while  a  fourth  proves  them  parallel,  etc. 

It  is  evident  then,  that  each  new  proposition  must  be 
proved  by  some  already  known  proposition  of  its  own 

203 


204  PRELIMINARY   CHAPTER 

class, — the  necessary  definitions  and  axioms  being  in- 
cluded in  their  respective  classes,  —  unless  it*  can  be 
proved  by  reasoning  from  its  converse  or  its  negative 
converse  (see  LOGIC,  p.  210).  As  an  illustration,  the 
first  proposition  in  solid  geometry  requiring  that  lines  be 
proved  parallel,  must  be  proved  by  some  plane  geom- 
etry method  of  proving  lines  parallel.  As  a  matter  of 
fact,  this  proposition  uses  the  definition  of  parallel  lines. 
The  second  proposition  requiring  that  lines  be  proved 
parallel  must  use  a  plane  geometry  method,  or  that  first 
proposition  of  solid  geometry,  and  so  successive  propositions 
build  up  a  solid  geometry  class  that  proves  lines  parallel. 
The  same  idea  holds  throughout  solid  geometry,  each 
proposition,  unless  pure  logic  is  sufficient,  being  proved 
by  a  plane  geometry  proposition  of  the  same  class,  or 
by  a  preceding  solid  geometry  proposition  of  that  class. 
In  case  a  proposition  is  the  first  one  of  a  new  class,  it 
uses  the  definitions  and  axioms  upon  which  that  class  is 
founded. 

When  a  proposition  has  been  classified,  and  the  pre- 
ceding propositions  of  that  class  have  been  thought  over, 
or  written  in  a  list,  the  given  conditions  should  be  care- 
fully considered.  The  possible  methods  of  proving  the 
proposition  that  do  not  seem  to  fit  the  conditions  should 
be  eliminated,  and  a  proof  should  be  attempted  by  which- 
ever of  the  remaining  methods  seems  the  best  for  the  pur- 
pose. In  many  propositions  each  of  the  methods  will 
give  a  proof.  After  the  method  of  proof  has  been  de- 
cided upon,  the  several  steps  are  found  by  repeated  appli- 
cations of  the  classification  method. 

The  use  of  this  method  of  analysis  is  more  easily  seen 
by  reference  to  the  following  illustrations,  which  are 
stated  in  condensed  form.  As  the  student  has  not  yet 


METHODS   OF   ATTACK  205 

studied  solid  geometry,  the  illustrations  are  taken  from 
plane  geometry. 

ILLUSTRATION  1.  The  opposite  sides  of  a  parallelo- 
gram are  equal. 

On  considering  the  methods  of  proving  sects  equal,  it 
appears  that  the  most  applicable  method  is  by  correspond- 
ing parts  of  congruent  triangles ;  therefore  a  diagonal 
should  be  drawn  to  form  the  triangles. 

On  considering  the  methods  of  proving  triangles  con- 
gruent, it  appears  that  it  is  impossible  to  use  a  theorem 
that  requires  two  or  three  pairs  of  equal  sides,  since  two 
pairs  of  the  sides  are  to  be  proved  equal.  Therefore  the 
congruence  of  the  triangles  must  be  proved  by  two  angles 
and  a  side  of  one  equal  to  the  corresponding  parts  of  the 
other. 

The  diagonal  is  the  common  side,  ancj.  the  use  of  the 
given  condition  (that  is,  the  fact  that  the  figure  is  a 
parallelogram,  and  the  definition  of  a  parallelogram) 
suggests  the  method  of  proving  the  angles  equal.  There 
are  two  ways  to  finish  the  proof,  (1)  by  using  two  angles 
and  the  included  side,  (2)  by  using  two  angles  and  the 
side  opposite  one  of  them. 

In  this  theorem  there  are  three  classifications,  that  is, 
at  three  points  it  must  be  decided  how  to  accomplish  the 
next  step,  and  in  each  step  the  examination  of  what  is 
wanted,  and  of  what  is  known,  shows  what  must  be  done. 

This  method  of  reasoning  is  not  confined  to  geometry, 
but  is  used  by  the  artisan,  the  engineer,  and  the  business 
man  who  must  meet  certain  conditions,  and  must  decide 
on  what  he  needs  to  do,  the  various  ^methods  of  doing  it,  and 
on  the  one  method  best  applicable  to  the  conditions  under 
which  he  is  working. 


206  PRELIMINARY   CHAPTER 

ILLUSTRATION  2.  If  a  triangle  is  inscribed  in  a  circle, 
the  product  of  two  of  its  sides  equals  the  altitude  to  the 
third  side  times  the  diameter  of  the  circle. 


Classification.  Products  can  be  proved  equal  by  means 
of  equivalent  areas,  or  by  a  proportion.  A  proportion 
can  be  proved  by  similar  triangles,  since  the  various 
methods  of  proving  a  proportion  can  usually  be  reduced 
to  the  method  by  similar  triangles.  Since  a  circle  is  given, 
it  can  be  used  to  measure  equal  angles  in  the  triangles 
that  are  to  be  proved  similar. 

So  far  the  classification  has  been  merely  mental  consid- 
eration of  the  required  conclusion  and  the  given  figure. 
Now  that  the  discussion  has  reached  the  point  of  deciding 
to  try  to  get  similar  triangles  by  the  use  of  angles,  the 
auxiliary  (or  construction)  lines  must  be  added  to  the 
figure.  Let  the  triangle  ABC,  with  the  altitude  CH  to  AB, 
be  inscribed  in  circle  O.  To  prove  BC  -  GA  —  CH  •  d,  or 

7?  C        C*fT 

—  = Since  CH  and  CA  are  in  the  triangle  CAH,  d 

d       (jA 

must  be  placed  so  as  to  be  in  a  triangle  witli  BC,  that 
is,  it  must  be  drawn  from  B  or  from  C.  In  this  figure 
C  is  chosen.  To  complete  the  triangle,  draw  the  sect  from 
7?  to  the  other  end  of  the  diameter  JT,  then  show  that  the 


METHODS   OF   ATTACK  207 

triangles  CAH  and  BCX  are  similar,  since  they  have  two 
angles  respectively  equal. 

Special  Case  of  Analysis  by  the  Classification  Method : 
Intersection  of  Loci.  Many  propositions  require  that  a 
locus  be  found  that  depends  on  two  or  three  loci  already 
known.  This  fact  is  discovered  by  the  examination  of 
the  required  conclusion ;  then  the  known  loci  are  con- 
structed. The  figure  formed  by  their  common  points, 
that  is,  the  intersection  of  the  loci,  is  the  required  locus. 

For  example  :  To  find  the  locus  of  points  equidistant 
from  two  given  points ;  and  also  equidistant  from  two 
given  intersecting  lines. 

The  locus  of  points  equidistant  from  two  given  points 
is  the  perpendicular  bisector  of  the  sect  joining  them, 
while  the  locus  of  points  equidistant  from  two  given  inter- 
secting lines  is  the  two  lines  bisecting  the  angles  between 
those  lines.  Therefore  the  required  locus  is  the  one  or 
more  points  in  which  the  perpendicular  bisector  meets 
the  bisectors  of  the  angles.  This  method  is  much  used 
in  solid  geometry.  Three  loci  are  sometimes  needed  to 
determine  the  required  locus. 

To  sum  up  the  classification  method  of  attack  : 

1.  Consider  the  known  methods  of  obtaining  the  re- 
quired conclusion. 

2.  Examine   those    methods  in   relation    to   the  given 
conditions,  and  eliminate   the    methods   that   seem   least 
applicable. 

3.  Of    the   methods   remaining   choose    the    one    that 
appears  most  applicable  to  reasoning  from  the  condition 
to  the  conclusion.     Classify  as  necessary  during  the  proof; 
draw  the   auxiliary  lines  that   the   method    used   makes 
necessary. 


208  PRELIMINARY  CHAPTER 

WARNING.  Remember  to  use  each  fact  of  the  given  conditions  in  the 
proof.  Since  the  conclusion  follows  only  from  the  conditions,  it  is  useless 
to  attempt  a  proof  that  does  not  make  use  of  all  conditions.  To  use  a  con- 
dition it  is  necessary  to  make  it  the  authority  for  at  least  one  step  of  the 
proof. 

The  Analysis  Method.  This  method  consists  in  consid- 
ering the  conclusion  to  be  true,  and  reasoning  from  the 
conclusion  to  the  conditions  upon  which  it  depends.  Then 
the  order  of  reasoning  is  reversed  to  prove  the  proposi- 
tion. This  method  is  of  most  use  in  constructions  that 
cannot  be  readily  done  by  the  classification  method. 

ILLUSTRATION.  To  draw  a  parallel  to  the  base  of  an 
isosceles  triangle  such  that  its  length  between  the  legs  of 
the  triangle  shall  equal  the  length  it  cuts  off  ( from  the 
base)  on  either  leg. 


A  B 

Suppose  that  XY  is  parallel  to  AB,  and  is  equal  to  XA 
and  to  BY.  Because  XY=  XA,  the  drawing  of  AY  forms 
an  isosceles  triangle,  and  Z  XAY  —  Z  XYA.  Because  XY 
is  parallel  to  AB,  Z  XYA  —  Z.  YAB.  That  is,  AY  bisects 
Z  A.  Hence  the  required  line  can  be  constructed  by 
bisecting  Z  A  by  A  Y,  and  drawing  a  line  from  Y  parallel 
to  AB. 

This  and  the  classification  method  are  evidently  varia- 
tions of  one  method.  In  this  example,  the  classification 
method  of  proof  would  be  :  XY  =  XA  if  Z  XAY  —  Z  XYA, 


METHODS  OF   ATTACK  209 

and  XT  II  AB  if  Z  XYA  —  Z  TAB,  both  of  which  conditions 
are  true  if  AY  bisects  /-A.  In  such  cases  as  this,  analysis 
is  probably  simpler. 

Algebraic  Analysis.  Many  propositions  and  exercises 
can  be  solved  by  algebraic  analysis.  They  are  usually  con- 
structions or  numerical  propositions.  The  general  method 
is  to  use  letters  for  the  unknown  parts,  and  then  to  solve 
algebraically  the  equations  that  can  be  written  in  terms  of 
these  letters.  If  the  question  is  a  construction,  the  re- 
quired element  is  constructed  after  it  has  been  found  in 
terms  of  the  given  parts  of  the  figure. 

ILLUSTRATIONS.  (1)  If  one  leg  of  a  right  triangle  is 
double  another  leg,  and  if  the  area  is  9  square  feet,  find 
the  lengths  of  the  sides. 

Let  the  shorter  leg  be  x,  the  longer  2  x,  then  J  x  •  2  x  =  9, 
and  the  legs  are  3  ft.  and  6  ft. 

(2)  To  construct  a  circle  whose  area  is  three  times 
that  of  a  given  circle. 

Call  the  radius  of  the  given  circle  r,  that  of  the  required  * 
circle  R.      Then  TrR2  =  3?rr2,  and  R  =  r V3.      Construct  R 
as  the  mean  proportional  between  r  and  3  r. 

(3)  Find  the  radius  of  a  circle  such  that  the  number 
of  linear  feet  in  the  circumference  equals  the  number 
of  square  feet  in  the  area. 

If  the  radius  is  r,  2  irr  =  Trr2,  and  r  =  2  feet. 

(4)  Find  by  algebraic  analysis  how  to  cut  a  given  sect 
in  mean  and  extreme  ratio. 

Let  the  given  sect  be  a,  and  the  required  mean  sect  be  x  ; 

then,  bv  definition  of  mean  and  extreme  ratio,  —  =  — -  — , 

x      a  —  x 

—  a  ±  a  V5 
or  x2  -f-  ax  —  a?  =  0.      On  solving  for  x,  x= 


210  PRELIMINARY   CHAPTER 

Using  the  positive   square  root   for   internal  division, 

(1)  construct  aV5  as  the  mean  proportional  between  a  and 
5  a  ;    (2)  subtract  a  from  «V5  ;    (3)  bisect  the  remainder  ; 
(4)  cut  the  resulting  sect  off  on  #,  and  a  will  be  cut  in 
mean  and  extreme  ratio. 

The  Use  of  Pure  Logic.1  For  convenience,  let  the  con- 
dition, or  hypothesis,  of  a  statement  be  represented  by  //, 
and,  the  conclusion  by  C. 

There  are  four  closely  related  conditional  statements,  as 
follows : 

(1)  If  IT  is  true,  then  C  is  true ; 

(2)  If  H  is  not  true,  then  C  is  not  true; 
(3)-  If  C  is  true,  then  n  is  true ; 

(4)  If  C  is  not  true,  then  H  is  not  true. 

Of  these,  (1)  and  (2)  are  negatives,  or  obverses,  of  each 
other,  or  either  is  said  to  be  the  negative  of  the  other. 

(1)  and  (3)  are  converses  of  each  other,  or  either  is  said 
to  be  the  converse  of  the  other.. 

(1)  and  (4)  are  negative  converses  of  each  other,  or  either 
is  said  to  be  the  negative  converse  of  the  other. 

Evidently  (3)  and  (4)  are  also  negatives  of  each  other, 

(2)  and  (3)  are  negative  converses,  etc. 

LAWS  CONCERNING  THE  TRUTH  OF  THE  RELATED 
STATEMENTS,  IF  ONE  OB  MOKE  STATEMENTS  ARE 
KNOWN  TO  BE  TRUE 

1.    Reasoning  from  a  Single  True  Statement. 

If  a  statement  is  true,  its  negative  converse  is  also  true;  its 
negative  and  its  converse  may,  or  may  not,  be  true. 

The  truth  of  this  law  can  be  seen  by  the  folio  wing  reason- 
ing :  Suppose  (1)  is  given,  that  is,  If  //is  true,  then  C  is 

1  See  also  Appendix,  §  318. 


METHODS   OF   ATTACK  211 

true.  Then  if  C  is  not  true,  either  (#)  H  is  true,  in  which 
case  (7  is  true,  or  (£>)  fl  is  not  true.  But  (a)  contradicts 
the  condition,  so  is  impossible,  and  (6)  must  follow,  that 
is,  the  negative  converse  of  a  true  statement  is  also  true. 

But  if  C  is  true,  H  may  be  either  true  or  untrue  without 
bringing  in  a  contradiction ;  and  if  H  is  untrue,  C  may  be 
true  or  untrue  without  bringing  in  a  contradiction,  so  the 
truth  or  untruth  of  a  converse  or  a  negative  cannot  be  es- 
tablished from  a  single  statement.  If  the  converse  or  the 
negative  is  true,  it  is  because  of  some  one  or  more  other 
propositions  used  in  connection  with  the  one  to  which  it  is 
related.  If  either  the  converse  or  the  negative  is  true, 
both  are  true,  for  they  are  negative  converses  of  each 
other.  .  See  also  Law  2. 

ILLUSTRATION.     Suppose  it  is  known  that 

(1)  If  two  lines  crossed  by  a  transversal  meet,  any 
pair  of  alternate  angles  are  unequal. 

Then  its  negative  converse  must  also  be  true,  that  is, 

(2)  If  a  pair  of  alternate  angles  formed  by  two  lines 
crossed  by  a  transversal  are  equal,  the  lines  are  parallel. 

Without  the  use  of  one  or  more  other  geometrical  state- 
ments, it  cannot  be  known  whether  (8),  its  converse,  or 
(4),  its  negative,  is  true  or  not. 

(3)  If  a  pair  of  alternate  angles  formed  by  two  lines 
crossed  by  a  transversal  are  unequal,  the  lines  meet  (are 
not  parallel}. 

(4)  If  two  lines  crossed  by  a  transversal  are  parallelf 
any  pair  of  alternate  angles  are  equal. 

To  Prove  the  Converse  of  a  True  Statement.  Show  that 
the  figure  constructed  by  using  the  condition  and  the 
figure  constructed  by  using  the  conclusion  are  identical. 
For  example,  to  prove  (4)  from  (2) ; 


212  PRELIMINARY  CHAPTER 

If  a  figure  is  drawn  with  the  angles  equal,  the  lines 
will  be  parallel  by  (2)  ;  if  the  lines  are  drawn  parallel, 
the  same  figure  will  result,  because  Through  a  point  there 
can  be  but  one  line  parallel  to  a  second  line,  which  is  the 
other  geometrical  statement  needed  in  this  proof. 

To  Prove  the  Negative  of  a  True  Statement.  Show  that 
the  figure  constructed  by  using  the  condition  and  the 
figure  constructed  by  using  the  conclusion  cannot  be  the 
same.  For  example,  to  prove  (3)  from  (2)  : 

If  a  figure  is  drawn  with  the  angles  equal,  the  lines  will  be 
parallel  by  (2)  ;  if  another  line  is  drawn  through  the  same 
point  making  the  angles  unequal,  it  will  be  a  different  line, 
and  so  cannot  be  parallel  since  there  can  be  but  one  parallel 
through  the  point.  To  prove  the  conclusion,  it  is  necessary 
to  use  the  additional  geometrical  principle  that  but  one 
parallel  to  a  given  line  can  be  drawn  through  a  given  point. 

WARNING.  It  must  not  be  assumed  that  the  converse  and  the  obverse 
of  a  proposition  are  even  likely  to  be  true  :  if  they  are  true,  they  can  be 
proved  as  in  the  examples  just  given,  or  by  the  method  of  reasoning  from 
statements  covering  all  possibilities,  which  will  be  given  later. 

NOTE.  It  should  be  observed  in  these  examples  of  the  methods 
used  in  proving  a  converse  or  a  negative,  that  a  line  is  added  to 
the  figure,  and  is  then  proved  to  coincide  or  not  to  coincide  with  one 
of  the  given  lines.  Nothing  is  assumed  in  constructing  it  as  to 
whether  or  not  it  is  the  same  line  as  the  one  in  the  figure,  but  if  it  is 
proved  to  be  the  same  line,  the  two  are  then  identical,  and  all  proper- 
ties of  the  one  are  also  properties  of  the  other. 

The  use  of  two  lines  which  may  be  identical  lines  should  not  be 
confused  with  the  use  of  two  given  lines,  for  all  given  elements  are 
supposed  to  be  distinct ;  as,  when  two  lines  are  given,  they  are  assumed 
to  be  different  lines.  Were  it  not  for  this  understanding,  the  proposi- 
tion, Lines  parallel  to  the  same  line  are  parallel  to  each  other,  would 
need  to  be  worded,  Two  different  lines  that  are  parallel  to  the  same  line 
are  parallel  to  each  other. 


METHODS  OF   ATTACK  213 

The  method  of  reasoning  from  the  negative  converse  is 
very  common  throughout  mathematics,  as  well  as  wherever 
reasoning  is  used  in  everyday  life.  The  most  common 
method  of  proving  a  proposition  or  other  fact  is  to  show  : 

(1)  That  a  certain  condition  must  give  a  certain  con- 
clusion (which  it  may  require  one  or  more  steps  of  reason- 
ing to  establish) ; 

(2)  That  the  conclusion  is  not  true. 

From  this  it  follows  by  the  negative  converse  law  that 
the  condition  could  not  have  been  true. 

A  few  examples  outside  of  mathematics  follow : 

1.  If  it  were  freezing,  the  puddles  would  be  covered  with 
ice.      There  is  no  ice  on  this  puddle,  so  it  is  not  freezing. 

2.  If  the  office  boy  had  been  attending  to  his  work,  the 
waste  paper  basket  would  have  been  emptied.     It  is  not 
emptied,  so  he  has  not  been  attending  to  his  work. 

3.  If  this  window  had  been  broken  from  the  inside,  the 
glass  would  have  fallen  out.     It  has  fallen  inside,  so  the 
glass  was  broken  from  the  outside. 

Other  examples  could  be  given  without  limit.  In 
mathematics,  when  several  steps  are  used  to  reason  from 
the  condition  to  an  untrue  conclusion,  and  so  to  show  that 
the  condition  must  have  been  untrue,  the  method  of  rea- 
soning is  sometimes  called  reductio  ad  absurdum. 

2.  Reasoning  from  True  Statements  Covering  all  Possi- 
bilities. 

If  statements  such  that  their  conditions  cover  all  possi- 
bilities are  true,  and  no  two  of  their  conclusions  can  be  true 
at  once,  then  their  converses  are  also  true. 

For  example,  suppose  it  has  been  proved  that 

(I)  If  two  sides  of  a  triangle  are  equal,  the  opposite 
angles  are  equal;  and 


214  PRELIMINARY  CHAPTER 

(2)  If  two  sides  of  a  triangle  are  unequal,  the  oppo- 
site angles  are  unequal,  the  larger  angle  being  opposite 
the  longer  side. 

Then,  since  the  conditions  with  regard  to  the  sides 
cover  all  possibilities,  and  only  one  of  the  conclusions  can 
be  true,  the  converses  of  these  statements  must  follow. 
Stated  with  regard  to  triangle  ABC, 

Given ;     If  BC  =  CA,  then  correspondingly  Z  A  ^  Z  B.    It 

follows  that  if  Z.A  $^B,  then  correspondingly  _B<7=  CA. 

If  the  explosion  of  the  battleship  Maine  was  within  the 
ship,  the  plates  must  have  been  blown  outward  ;  if  it  was 
outside  the  ship,  the  plates  must  have  been  blown  inward  ; 
then  the  converses  of  these  statements  are  also  true,  and 
since  it  was  found  that  the  plates  were  bent  in,  the  explo- 
sion must  have  been  outside  the  ship. 

The  truth  of  converse  statements  may  also  be  shown  by 
elimination  of  possibilities,  that  is  by  showing  that  all  but 
one  of  the  possibilities  are  untrue,  and  that  the  one  remain- 
ing is  therefore  the  true  one  because  it  is  the  only  remain- 
ing possibility.  For  instance,  in  the  foregoing  geometrical 
illustration,  given  the  same  conditions,  to  find  what  is  true 
if  Z.  A  >  Z  B. 

If  BC  were  equal  to  CA,  then  Z  A  would  be  equal  to 
/.  B,  which  is  not  true. 

If  BC  were  <  CA,  then  Z.  A  would  be  <Z.B,  which  is 
not  true ;  therefore  BC  is  >  CA,  because  that  is  the  only 
remaining  possibility. 

Order  and  Arrangement  of  a  Proof.  In  writing  out  a 
formal  proof,  it  is  best  to  have  a  definite  order  of  arrange- 
ment that  will  show  all  the  important  details  of  the  proof. 
The  following  arrangement  is  recommended  for  all  propo- 
sitions: 


METHODS   OF   ATTACK  215 

- 

STATEMENT  OF  PROPOSITION 

Diagram 

Given.     The  conditions  of  the  proposition. 

To  prove.  The  conclusion.  (Both  condition  and  con- 
clusion should  be  in  terms  of  the  figure.  If  it  is  a  con- 
struction, Required  should  take  the  place  of  To  prove.) 

Proof.  1.  The  proof  should  follow  in  numbered  steps. 
(If  it  is  a  construction,  the  proof  may  be  divided  into 
Construction  and  Proof.) 

ILLUSTRATION 
A  line  and  a  point  outside  determine  a  plane. 


Given.     Line  AB  and  external  point  P. 

To  prove.     AB  and  P  determine  a  plane. 

Proof.     1.    Let  K  and  L  be  two  points  on  AB,  then  K,  L, 

and  P   determine  a  plane   (definition  of 

plane,  §  5,  p.  233). 

2.  But  AB  is  entirely  in  this  plane,  for  two  of 

its  points  are  in  the  plane  (axiom  of  line 
and  plane,  §  6,  p.  234). 

3.  Therefore  AB  and  P  determine  a  plane,  for 

they  lie  in  one,  and  but  one,  plane. 


SECTION   III.     THE   REPRESENTATION   OF  SOLID 
GEOMETRY   FIGURES 

Since  in  solid  geometry  three-dimensional  figures  are 
usually  represented  on  the  paper  or  on  the  blackboard,  that 
is,  on  a  surface  of  but  two  dimensions,  the  student  needs 
some  idea  of  how  to  make  the  figures  appear  solid.  While 
there  is  neither  space  nor  time  for  a  study  of  drawing,  it 
is  believed  that  the  following  brief  directions  may  be  of 
assistance  in  giving  this  idea. 

Unless  the  student  has  given  some  time  to  the  study  of 
perspective,  the  parallel  line  method  will  be  the  simplest 
for  him  to  use.  In  this,  a  plane,  or  rather,  —  since  a  plane 
is  unlimited  in  extent,  —  that  portion  of  it  shown  in  the 
figure,  is,  in  general,  represented  by  a  parallelogram.  It 
follows  that  many  solids,  since  they  are  bounded  by  planes, 
can  be  drawn  by  the  use  of  sets  of  parallel  lines.  This, 
of  course,  is  not  true  of  solids  the  conditions  of  which 
require  that  some  portion  of  a  plane  shown  in  the  figure 
shall  be  a  circle,  or  a  polygon  other  than  a  parallelogram. 
Where  parallel  lines  are  used,  one  set  is  usually  drawn 
horizontally,  the  other  somewhat  inclined  to  the  vertical. 

There  are  a  few  general  rules  that  are  of  assistance  in 
drawing  all  the  figures,  the  most  important  being  the 
following : 

The  figures  are  usually  supposed  to  be  viewed  from 
a  point  a  little  above,  and  either  directly  in  front,  or  a 
little  to  one  side. 

216 


REPRESENTATION  OF  SOLIDS  217 

*•*» 

Lines  nearer  to  the  observer  should  be  somewhat  heavier 
than  those  at  the  back  of  the  figure. 

As  planes  are  supposed  to  be  opaque,  all  lines  appar- 
ently covered  by  them  should  be  left  out  entirely,  or,  if 
needed  in  the  figure,  should  be  dotted.  Where  it  seems 
desirable,  lines  that  should  not  appear  in  the  figure  may 
be  drawn  and  afterwards  erased. 

A  circle,  unless  viewed  from  a  point  directly  opposite 
the  center,  appears  elliptical,  the  ellipse  being  narrower 
as  the  eye  is  nearer  its  plane. 

It  should  be  noted  that  two  lines  that  intersect  in  a 
drawing  do  not  necessarily  intersect  in  the  three-dimen- 
sional figure,  and  that  the  actual  length  of  a  line,  or  the 
size  of  an  angle,  may  be  very  different  from  its  representa- 
tion, as  each  part  is  drawn  so  as  to  appear  to  be  of  the 
right  size  in  the  three-dimensional  figure  represented. 

The  Use  of  Squared  Paper.  At  the  beginning,  squared 
paper  will  be  found  very  convenient  for  drawing  the  solid 
geometry  figures,  for  with  it  equal  lines,  parallel  lines, 
and  perpendicular  lines  can  be  drawn  with  no  instru- 
ment but  the  straightedge.  As  the  student  gains  skill  in 
drawing  he  should  not  confine  himself  to  this  method, 
but  should  draw  accurate  figures  on  unruled  paper,  using 
compasses  and  ruler.  He  should  also  draw  reasonably 
accurate  freehand  figures.  If  models  of  the  solid  figures 
are  used  in  addition  to  the  drawings,  the  two  together 
will  give  the  best  possible  idea  of  the  three-dimensional 
figures. 

The  following  explanation  shows  how  to  draw  parallel 
or  perpendicular  lines  on  squared  paper: 

(1)  Parallel  Lines.  (Fig.  1).  Draw  the  lines  so  that 
they  connect  points  the  same  distance  apart,  measured 


218  PRELIMINARY   CHAPTER 

along  the  lines  of  the  paper,  to  the  right  or  left,  and  up 
or  down.  For  example,  in  Fig.  1,  to  draw  a  line  from  C 
parallel  to  AB,  note  that  AB  extends  to  the  right  5  spaces, 
and  up  15  spaces ;  therefore  count  from  C  to  the  right  5 
spaces,  and  up  15  spaces  to  D,  and  draw  CD.  This  also 
makes  the  lines  equal,  as  can  be  seen  by  this  illustration, 
where  CD  =  AB.  If  the  lines  are  not  to  be  of  the  same 
length,  the  same  method  is  used,  and  that  part  of  CD 
that  is  needed  is  taken.  - 

(2)  Perpendicular  Lines.  (Fig.  2).  Count  as  before, 
but  draw  the  new  line  so  that  one  direction  is  reversed,  — 
that  is,  up  for  down,  left  for  right,  etc.,  —  and  so  that  the 
two  numbers  are  interchanged.  In  Fig.  2,  to  draw  a  line 
from  P  perpendicular  to  AB,  note  that  AB  extends  10 
spaces  down,  and  15  spaces  to  the  right,  so  draw  from  P 
to  a  point  O,  15  spaces  down,  and  10  spaces  to  the  left,  or, 
15  spaces  up,  and  10  spaces  to  the  right. 

The  Plane  (Figs.  3,  4).  Draw  a  parallelogram,  usually 
with  two  edges  horizontal.  The  other  positions  in  which 
a  plane  most  frequently  appears  will  be  taken  up  in  the 
following  figures.  If,  for  any  reason,  it  is  desirable  to 
emphasize  the  fact  that  the  plane  is  unlimited  in  extent, 
one  pair  of  edges  can  be  broken,  as  in  Fig.  4.  • 

Two  Lines  in  Space  (Figs.  5,  8).  Intersecting  or  par- 
allel lines  do  not  differ  from  those  in  plane  geometry. 
Two  lines  not  in  the  same  plane,  called  non-coplanar  lines, 
cannot  be  distinguished  from  intersecting  lines,  unless 
there  is  more  in  the  figure  to  help  show  their  relative 
position.  In  Fig.  5,  the  plane  M  appears  to  contain  CD, 
and  so  makes  it  appear  that  AB  and  CD  are  not  in  the 
same  plane.  In  P'ig.  8,  AB  and  ES  are  not  in  the  same 


REPRESENTATION   OF   SOLIDS 


219 


220 


PRELIMINARY   CHAPTER 


PLATE  II 


F/G.  3.       PLANES          FIG.  4 


F/G.  5.  TWO  NON-COPLAA/AR  L/MES 


F/G.  6.  A  L/NE  AND  A  PLANE  F/G.  7 


REPRESENTATION    OF   SOLIDS  221 

plane,  as  is  shown  by  the  points  in  which  they  meet  the 
planes  M,  N,  and  P. 

A  Line  and  a  Plane  (Figs.  5,  6,  7,  8,  9). 

(1)  The  Line  in  the  Plane.     Draw  it  within  the  paral- 
lelogram (or  other  outline)  that  represents  the  plane,  as 
AC  and  OB  in  Fig.  6. 

(2)  The  Line  Parallel  to  the  Plane.     Draw  it  parallel 
to  a  pair  of  sides  of  the  parallelogram,  preferably  to  the 
horizontal  sides,  as  ES  in  Fig.  6.     If  more  than  one  line 
is  to  be  drawn  through  a  point  parallel  to  the  plane,  draw 
them  so  that  they  will  be  within  the  outline  that  repre- 
sents another  plane  parallel  to  the  given  plane.     Thus,  in 
Fig.  8,  OK  and  OL  are  parallel  to  plane  P. 

(3)  The   Line   Perpendicular   to   the   Plane.      Draw   it 
perpendicular  to  the  pair    of   horizontal  sides,  as  PO  in 
Fig.  6,  AB  in  Fig.  8,  and  PO  in  Fig.  9.     If  a  portion  of 
a  plane  is  represented  by  an  outline  other  than  a  parallel- 
ogram, imagine  the  parallelogram  you  would  draw  to  rep- 
resent that  plane,  and    then    draw  the  line  so  as  to  be 
perpendicular  to  two  sides  of  that   parallelogram.      See 
P'o'  in  Fig.  7. 

(4)  The  Line  Oblique  to  the  Plane.     Draw  it  so  as  not 
to  be  perpendicular  to  the  horizontal  sides,  as  PA,  PB, 
PC  in  Fig.  6,  BC  and  ES  in  Fig.  8,  and  PB  in  Fig.  9. 

Parallel  Planes  (Fig.  8).  Represent  them  by  paral- 
lelograms with  their  corresponding  sides  parallel.  To 
represent  portions  of  parallel  planes,  if  the  conditions  of 
the  figure  require  some  other  outline,  as  in  Fig.  22,  the 
corresponding  sides  should  still  be  parallel. 

Intersecting  Planes  (Figs.  9, 10).  First  draw  the  line  of 
intersection,  preferably  somewhat  oblique  to  the  vertical, 


222 


PRELIMINARY   CHAPTER 


PLATE    III 


£ 


5   C 

8.  PARALLEL  PLAMES 


f/G.  9.  /MTEPSECr/A/G  £>LAA/ES 


*      REPRESENTATION   OF   SOLIDS  223 

so  that  the  observer  will  appear  to  be  looking  at  one 
plane  from  a  little  above,  and  at  the  other  from  a  little 
to  one  side.  Then  fill  in  the  outlines  of  the  planes,  keep- 
ing one  pair  of  sides  of  the  parallelogram  representing 
each  plane  parallel  to  the  intersection.  If  the  planes  are 
to  be  perpendicular  to  each  other,  make  a  pair  of  sides  of 
one  parallelogram  perpendicular  to  a  pair  of  sides  of  the 
other,  as  in  Fig.  10;  otherwise,  let  the  sides  not  parallel  to 
the  intersection  be  oblique  to  each  other,  as  in  Fig.  9. 

In  Fig.  9,  the  plane  PAO  is  perpendicular  to  the  plane 
M  and  to  the  plane  N,  but,  on  account  of  its  position  with 
reference  to  those  two  planes,  it  cannot  be  shown  in  the 
same  way  as  in  Fig.  10.  It  is  made  to  appear  perpen- 
dicular by  being  drawn  so  as  to  appear  parallel  to  the 
horizontal  sides  of  the  parallelogram  representing  plane 
If,  and  to  the  corresponding  sides  of  the  parallelogram 
representing  plane  N.  Note  that  PA  and  AO  are  made 
to  appear  perpendicular  to  the  intersection  by  being 
drawn  parallel  to  a  pair  of  edges.  In  the  same  way,  in 
Fig.  10,  PO  and  OA  are  perpendicular  to  the  intersection, 
while  OB  is  oblique  to  it. 

Three  Planes  Intersecting  in  Parallel  Lines  (Figs. 
11,  12).  Draw  the  parallel  intersections,  dotting  what- 
ever is  covered  by  the  planes,  as  in  Fig.  12.  The  planes 
can  be  represented  by  completing  the  parallelograms  as  in 
these  figures,  or  by  using  a  broken  edge,  as  in  Fig.  13. 

For  more  than  three  planes  intersecting  in  parallel 
lines,  see  Figs.  17,  18,  19. 

Three  or  more  Planes  Concurrent  in  a  Point  (Figs. 
13,  14).  Choose  the  point  that  is  to  be  the  vertex,  as  F, 
and  draw  the  edges,  and  fill  in  the  other  boundaries 


224 


PRELIMINARY   CHAPTER 


PLATE  W 


F/G.  /O.  P&*P£N£)/C(/LAa 


FIG.  /2 


FIG.  // 


THREE  PLANES 

PARALLEL  L//VE-S 


/A/ 


REPRESENTATION   OF   SOLIDS 


225 


/7G.  /3 

Th 'PEE OA MOPE PZ.A/VES 
///A  PO//VT 


A 


/70./6 


T/fAfE  OR  MORE  PLANES  MEET/A/G  /A/ 

A  COMMO/V  L/ME 


226  PRELIMINARY   CHAPTER 

afterwards,  or  draw  the  other  boundary,  and  connect 
its  vertices  to  the  vertex  F.  Be  careful  that  the  edges 
behind  the  planes,  as  VE  and  VF  in  Fig.  14,  do  not  lie  so 
near  the  other  edges  as  to  cause  confusion.  The  termina- 
tion of  the  figure  can  be  by  a  plane,  as  ABC  in  Fig.  13, 
or  ABCDEF  in  Fig.  14,  or  by  a  broken  boundary,  as 'in 
Fig.  13,  when  the  fact  that  the  figure  is  unlimited  is  to 
be  emphasized. 

Three  or  more  Planes  Meeting  in  a  Common  Line  (Figs. 
15,  16).  Draw  the  intersection,  then  the  plane  nearest 
the  observer,  and  so  on  ;  keep  one  side  parallel  to  the 
intersection.  The  hidden  lines  can  be  dotted  when  they 
are  needed  in  the  figure. 

Parallelepipeds  and  other  Prisms  (Figs.  17,  18,  19). 
Draw  one  base,  then  draw  parallel  equal  edges,  and  fill  in 
the  other  base  last.  A  pentagonal  prism  is  shown  in  Fig. 
17,  an  oblique  parallelepiped  in  Fig.  18,  and  a  rectangular 
parallelepiped  in  Fig.  19. 

The  Cylinder  (Fig.  20).  Draw  two  parallel  equal 
lines  to  represent  two  elements,  then  draw  ellipses  between 
their  ends.  ABCD  is  a  section  of  the  cylinder  through  an 
element. 

The  Pyramid  (Figs.  13,  14,  21,  22).  The  figures  v- 
ABC  in  Fig.  13,  and  V- ABCDEF  in  Fig.  14  are  pyramids 
and  can  be  drawn  as  explained  there.  In  Fig.  21  the 
pyramid  is  regular,  as  is  indicated  by  its  regular  base,  and 
by  the  fact  that  V  is  on  the  perpendicular  at  the  center  of 
the  base. 


REPRESENTATION   OF   SOLIDS 


227 


PLATE  VI 


Fid  17          f/G.  /8  F/G.  /9 

PARALLELEP/PEOS  A/VD  Or/Y£& 


A 

F/G.  20 
CYL/MDER 


A 
B 

F/G.2/          F/G.  2 2 
PYRAM/DS 


228  PRELIMINARY  CHAPTER 

For  a  frustum,  draw  a  pyramid,  with  the  lateral  edges 
filled  in  lightly  so  that  they  can  be  erased  where  they  are 
not  needed  ;  then  from  some  point  on  one  edge,  as  P  in 
Fig.  22,  draw  a  parallel  section  by  drawing  lines  parallel 
to  the  corresponding  edges  of  the  base  from  each  lateral 
edge  to  the  next  lateral  edge.  For  example,  PQ  is  paral- 
lel to  J?<7,  etc.  The  parts  of  the  lateral  edges  from  the 
upper  base  to  the  vertex  should  then  be  erased. 

The  Cone  (Figs.  23,  24,  25).  A  cone  can  be  drawn  by 
drawing  two  lines  from  the  point  to  be  used  as  its  vertex, 
and  then  filling  in  the  elliptical  base.  Figure  23  is  an 
oblique  cone  with  a  triangular  section  ABV ;  Fig.  24  is  a 
right  cone  with  its  altitude  meeting  the  base  at  its  center. 

In  drawing  a  frustum,  draw  two  elements  between  par- 
allel lines,  that  is,  so  that  they  would  be  legs  of  a  trapezoid 
if  their  ends  were  connected,  then  fill  in  the  elliptical 
bases,  as  in  Fig.  25. 

The  Sphere  (Figs.  26,  27,  28,  29).  The  sphere  is  rep- 
resented by  a  circle,  the  appearance  of  solidity  being  given 
by  the  other  lines  drawn  in  the  figure.  Of  these,  circles  on 
the  surface  of  the  sphere  are  the  most  frequent.  In  Fig.  26 
a  great  circle  G  and  a  small  circle  8  are  shown,  with  their 
radii,  and,  since  they  are  parallel,  with  their  common  axis 
PPf.  All  circles  except  the  one  used  for  the  outline  of  the 
sphere  should  appear  as  ellipses,  the  ellipse  being  narrower 
as  it  is  supposed  to  be  looked  at  from  a  point  more  nearly 
in  its  plane,  and  more  like  a  circle  as  it  is  supposed  to  be 
looked  at  from  a  point  more  nearly  on  its  axis. 

To  draw  the  arc  of  'a  great  circle,  the  arc  should  be  so 
curved  that  it  will  appear  to  meet  the  outline  circle  at  the 
ends  of  a  diameter.  In  Fig.  27  the  spherical  triangle 


REPRESENTATION   OF   SOLIDS 


229 


PLATE  VII 


FIG.  ^5 


230  PRELIMINARY  CHAPTER 

ABC  is  drawn,  and  the  sides  are  continued  in  dotted  lines, 
to  show  how  their  circles  would  appear  to  pass  through 
the  ends  of  diameters.  In  this  figure,  the  observer  is  look- 
ing toward  the  center  of  the  sphere  from  a  point  such  that 
the  line  along  which  he  is  looking  meets  the  surface  in- 
side the  triangle,  as  is  shown  by  the  fact  that  the  sides 
all  curve  away  from  the  inside  of  the  triangle.  In  Fig. 
28  the  eye  is  in  such  a  position  that  the  sides  of  the  tri- 
angle all  appear  to  curve  away  from  a  point  outside  of  the 
triangle.  It  makes  little  difference  in  what  position  the 
eye  is  supposed  to  be,  if  the  arcs  are  drawn  through 
the  ends  of  diameters  so  that  they  give  the  appearance  of 
solidity  to  the  sphere. 

To  draw  polar  triangles,  draw  the  axis  of  each  of  the 
circles  that  form  one  triangle,  using  only  that  half  of  the 
axis  that  meets  the  same  hemispherical  surface  on  which 
the  opposite  vertex  lies,  and  draw  great  circle  arcs 
through  the  poles  found,  using  the  method  shown  in  Figs. 
27  and  28. 

Other  Figures.  The  diagrams  shown  are  those  most  fre- 
quently used  in  solid  geometry,  the  others  of  importance 
being  for  the  most  part  combinations  of  these.  For  ex- 
ample, polyhedral  angles  are  formed  by  three  or  more 
planes  meeting  in  a  point ;  a  spherical  pyramid  is  a  com- 
bination of  the  pyramid  with  great  circle  arcs,  the  regular 
tetrahedron  and  the  regular  octahedron  are  respectively 
a  pyramid,  and  two  pyramids  placed  base  to  base,  while 
a  regular  hexahedron  is  a  rectangular  parallelepiped  with 
equal  edges.  If  the  pupil  has  clearly  in  mind  the  few 
principles  by  which  these  diagrams  have  been  drawn,  the 
building  up  of  new  and  more  complicated  diagrams  should 
not  be  found  difficult. 


REPRESENTATION  OF   SOLIDS 


231 


PLATE  Vffl 


F/o.  28.  SPHER/CAL  TR/AMGLES 


F/o.29.     POLAR  TR/AMGLES 


232  PRELIMINARY  CHAPTER 

NOTES 

Statements  marked  with  an  asterisk  require  proof. 
Such  statements  include  many  simple  deductions  from  the 
definitions  and  axioms,  not  important  enough  to  receive 
the  emphasis  given  to  propositions,  as  well  as  other  im- 
portant statements  that  follow  so  naturally  from  the  dis- 
cussion that  a  more  or  less  informal  proof  of  them  seems 
most  satisfactory.  Both  these  classes  of  statements,  which 
have  been  italicized,  may  be  considered  as  corollaries  of 
the  definitions  and  axioms. 

In  order  to  avoid  circumlocution  and  monotonous  repe- 
tition in  the  statements  concerning  mensuration,  this 
book,  in  speaking  of  the  length  of  a  line,  usually  omits 
"the  length  of,"  and  in  speaking  of  the  areas  of  plane 
figures  usually  omits  "the  area  of,"  wherever  there  is 
no  doubt  as  to  the  meaning.  For  example,  in  the  state- 
ment, "  The  volume  of  a  parallelepiped  equals  the  product 
of  its  base  and  its  altitude"  "base"  manifestly  means  "the 
area  of  the  base  "  and,  "  altitude,"  "  the  length  of  the  alti- 
tude," and  should  be  so  interpreted. 


BOOK   VI.     LINES   AND   PLANES 

SECTION   I.     THE   PLANE 

1.  Space.     The  space  in  which  everything  exists  is,  as 
far  as  experience  shows,  unlimited.     It  is  also  divisible, 
for  all  the  bodies  with  which  we  are  familiar  occupy  por- 
tions of  space.     While  the  space  studied  in  solid  geom- 
etry  (sometimes  called   Euclidean  space)  appears  to  be 
the  same  as  that  in  which  we  exist,  it  is  assumed  only  to 
be  a  space  such  that  in  it  the  axioms  and  postulates  of 
geometry  are  true.      Euclidean  space   is  assumed  to  be 
unlimited  in  extent  and  to  be  divisible. 

2.  Solids.     Any  limited   portion  of   space   is  called  a 
geometric  solid,  or  simply  a  solid.     The  term  "  solid  "  must 
not  be  confused  with  "  solid  body,"  for  the  geometric  solid 
is  studied  as  a  part  of  space,  and  is  entirely  irrespective 
of  any  physical  body  that  might  occupy  it. 

3.  Surfaces.     That   which    separates    one    portion    of 
space  from  an  adjoining  portion  is  called  a  surface.     A 
surface  may  be  limited  or  unlimited  in  extent,  and  can 
have  limited  portions. 

4.  Determining  a  Surface.      As  in   plane   geometry  a 
straight  line  is  said  to  be  determined  by  any  two  of  its 
points,  so  in   solid   geometry   a  surface  of  any  specified 
kind  is  said  to  be  determined  by  given  points,  lines,  etc., 
if  it  is  the  only  surface  of  its  kind  that  contains  them. 

5.  Planes.     The   surface   that   is   determined   by   any 
three  of  its  points  that  are  not  in  a  straight  line  is  called 

233 


234  LINES   AND  PLANES 

a  plane  surface,  or  simply  a  plane.  A  plane  is  unlimited 
in  .extent,  but  limited  portions  of  a  plane  can  be  consid- 
ered. Familiar  examples  of  a  portion  of  a  plane  are,  the 
surface  of  a  desk,  the  surface  of  a  blackboard,  or  any  sur- 
face commonly  spoken  of  as  "  flat." 

6.  Axiom  of  the  Straight  Line  and  the  Plane.     If  two 

points  of  a  straight  line  are  in  a  plane,  the  whole  line 
is  in  the  plane. 

7.  Similarity  between  the  Plane  and  the  Straight  Line. 

The  plane  occupies  the  same  position  among  surfaces  that 
the  straight  line  does  among  lines;  as  indicated  by  the 
axiom,  it  is  straight,  — that  is,  a  straight  line  could  lie  in 
it, — through  any  two  of  its  points.  Another  similarity 
arises  from  the  fact  that  the  straight  line  is  the  one  line 
determined  by  two  of  its  points,  while  the  plane  is  the  one 
surface  determined  by  three  of  its  points. 

8.  Solid   Geometry.     Solid   geometry  treats  of   figures 
that  do  not  lie  entirely  in  one  plane. 

In  proving  the  propositions  of  solid  geometry,  any  of 
the  definitions,  axioms,  and  propositions  of  plane  geom- 
etry may  be  used,  as  well  as  the  definitions  and  axioms  of 
solid  geometry.  It  is  therefore  necessary  to  find  out  what 
planes  exist  in  a  figure,  and  what  points  and  lines  are  in 
those  planes,  in  order  to  be  able  to  apply  plane  geometry 
to  as  great  an  extent  as  possible,  and  to  guard  against 
using  plane  geometry  propositions  where  they  do  not  hold. 

9.  Determining  a  Plane.     To   determine   a  plane,  two 
facts  must  be  proved:  that  the  determining  points  or  lines 
are  such  that  they  lie  wholly  in  that  plane,  and  that  no 
other  plane  can  contain  them.     It  has  already  been  said 
that  a  plane  is  determined  by 

(1)    three  points  not  in  a  straight  line; 


THE  PLANE  235 

It  is  also  determined  by 

(2)  A.  line  and  a  point  outside  that  line  ; 

for  two  points  on  the  line  and  the  point  outside  determine 
the  plane,  and  the  line  is  wholly  in  the  plane  since  two  of 
its  points  are  in  the  plane  ; 

(3)  two  intersecting  lines  (Prove  as  in  (2))  ; 

(4)  two  parallel   lines  (See  the   definition  of   parallel 
lines). 

10.   Representation   of  a  Plane.     Although   a   plane   is 
unlimited  in  extent,  only  a  limited  part  can  be   shown 
in  the  drawing  of  a  figure,  —  just  as  in  plane  geometry 
only   a  limited  part  of   an 
unlimited    line    is    shown. 
A  parallelogram   has  been 
found  to  be,  in  general,  the 
best  representation  to  use 

in  such  a  drawing.     It   is          

well  to  make  the  side  sup- 
posed to  be  nearer  the  observer  a  little  heavier  than  the 
others.  Such  a  plane  is  denoted  by  a  single  letter,  as  plane 
M,  by  the  Betters  at  opposite  vertices  of  the  parallelogram,  as 
plane  AB,  or  by  some  determining  points,  or  lines,  as  plane 
ABC,  if»it  contains  points  A,  B,  and  C,  or  plane  «,  b  if  it 
contains  lines  a  and  b.  For  a  more  extended  discussion 
of  figures,  see  Section  III  of  the  Preliminary  Chapter. 

EXERCISES 

1.  Does  a  triangle  determine  a  plane ?     Why? 

2.  Why  does  a  three-legged  stool  stand  firmly  on  the  floor,  while  a 
four-legged  chair  or  table  will  sometimes  rock  ? 

3.  If  four  points  are  given,  how  many  planes  can  be  determined? 
What  is  the  smallest  number  of  planes  in  which  the  sides  of  a  quadri- 
lateral can  lie  ?    In  how  many  planes  can  all  the  sides  of  a  parallelo- 
gram lie? 


236  LINES  AND  PLANES 

4.  In  a  series  of  parallels  such  that  not  more  than  two  lie  in  any 
one  plane,  how  many  planes  would  be  determined  by  using  three  of 
them  ?    four  ? 

5.  Show  that  all  transversals  of  two  intersecting  lines  are  in  the 
same  plane. 

11.  Coplanar  Figures.  Figures  in  the  same  plane  — 
such  as  two  intersecting  lines  —  are  said  to  be  coplanar, 
those  not  in  the  same  plane  are  said  to  be  non-coplanar. 
When  figures  are  said  to  lie  in  the  same  plane,  or  to  be 
coplanar,  it  is  meant  that  a  plane  can  be  passed  through 
the  figures  so  as  to  entirely  contain  them,  —  not  that 
there  is  necessarily  a  given  plane  that  does  contain  them. 
Similarly,  when  any  figures  are  said  to  be  non-coplanar, 
the  meaning  is  that  no  plane  could  be  passed  so  as  to  contain 
them.  The  fact  that  two  lines,  or  other  parts,  appear  in 
different  planes  of  a  given  figure  does  not,  therefore,  mean 
that  they  must  be  non-coplanar,  for  there  may  be  some 
other  plane  that  can  be  passed  through  them. 


SECTION  II.     RELATIVE  POSITIONS  OF   LINES  AND 

PLANES  IN  SPACE ;    PARALLELS 

AND  INTERSECTIONS 

12.  Necessity  of  the  Study  of  Relative  Positions.     In 

plane  geometry  the  study  of  the  relative  positions  of 
points  and  lines  is  very  informal,  because  the  possibilities 
are  few ;  but  in  solid  geometry,  points,  lines,  and  planes 
admit  of  many  relative  positions.  In  order  to  give  a 
clear  idea  of  the  figures,  their  more  fundamental  combi- 
nations will  be  studied.  ^ 

13.  Relative  Positions  of  Two  Lines.     Two  lines  can  be 

(1)  Coplanar,  therefore,  as  in  plane  geometry ; 
(a)  intersecting. 

(6)  parallel. 

(2)  Non-coplanar,    therefore   neither    intersecting    nor 
parallel. 

An  illustration  of  non-coplanar  lines  can  be  shown  by 
crossing  two  pencils,  and  then  moving  one  away  from  the 
other  without  changing  the  position  of  either  except  for 
this  separation.  The  edge  of  the  side  wall  and  the  ceiling 
of  a  room,  and  the  edge  of  the  back  wall  and  the  floor 
would  also  be  a  pair  of  such  lines. 

The  angle  between  two  non-coplanar  lines  means  the  angle 
between  one  of  them  and  an  intersecting  line  parallel  to 
the  other.  All  such  angles  formed  for  a  certain  pair  of 
lines  are  equal,  as  is  proved  in  §  42. 

When  two  lines  are  neither  intersecting  nor  parallel, 
they  are  not  coplanar.  When  two  lines  are  given  without 

237 


238  LINES   AND   PLANES 

any  conditions  that  show  without  further  proof  whether 
or  not  they  are  intersecting  or  parallel,  they  cannot  be 
assumed  to  determine  a  plane.  One  of  them  with  a  point 
of  the  other  does,  however,  determine  a  plane,  and  if  the 
second  line  should  also  lie  in  that  plane,  it  may  then  be 
possible  to  prove  this  fact. 

6.  Transversals  of  two  non-coplanar  lines  are  neither  parallel  nor 
concurrent  unless  they  meet  at  a  point  on  one  of  the  lines. 

14.  Relative  Positions  of  a  Line  and  a  Plane.     A  line 
either  (1)  lies  in  a  given  plane  ;   (2)  intersects  the  plane  ; 
or  (3)  is  parallel  to  the  plane. 

A  line  has  already  been  said  to  lie  in  a  certain  plane  if 
two  of  its  points  are  in  that  plane.  Other  methods  of 
proving  ihat  it  lies  in  a  certain  plane  will  be  taken  up  later. 

A  line  intersects  a  plane  if  it  meets  the  plane  in  one 
point  only.  The  point  is  called  the  foot  of  the  line  with 
respect  to  that  plane. 

A  line  is  parallel  to  a  plane,  and  the  plane  is  parallel  to 
the  line,  if  they  have  no  point  in  common.  The  given 
line  evidently  cannot  meet  any  line  in  the  plane,  but  it  is 
parallel  only  to  such  of  those  lines  as  are  coplanar  with  it. 
Why? 

15.  Relative  Positions  of  Two  Planes.    Two  planes  either 
(1)   coincide;    (2)  intersect ;    or   (3)   are   parallel.     The 
study  of  two  planes  will  also  include  some  use  of  a  line 
with  the  two  planes. 

16.  Planes  coincide  if  no  point  of  one  is  outside  the 
other.     Coincidence  is  proved  by  showing  the  two  planes 
to  be  determined  by  the  same  determining  points," lines,  or 
points  and  lines.      (See  §  9.) 

17.  Two  planes  intersect  when  they  have  at  least  one 
common  point,  but  do  not  coincide* 


RELATIVE   POSITIONS   OF   LINES   AND   PLANES     239 

18.  Axiom    on   Intersecting    Planes.     Two  intersecting 
planes  have  at  least  two  points  in  common. 

19.  Theorem   I.      The  intersection  of  two  planes  is  a 
straight  line. 

They  intersect  in  two  points.  What  is  known  of  the 
line  through  those  points?  Can  the  two  planes  have  any 
other  points  in  common  ? 

7.  Three  planes  that  do  not  contain  the  same  straight  line  can 
have  but  one  point  in  common. 

8.  There  can  be  but  one  line  from  an  external  point  cutting  two 
non-coplanar  lines. 

9.  If  a  paper  is  folded  and  creased  smoothly,  why  does  it  form  a 
straight  edge  along  the  fold? 

20.  COR.       If  a   line   is   parallel  to   a   plane,   it   is 
parallel  to  the  intersection  of  any  plane  in  which  it  lies, 
with  that  plane. 

10.  If  a  line  in  one  of  two  intersecting  planes  is  parallel  to  the 
other  plane,  it  is  parallel  to  the  intersection. 

11.  If  a  line  is  parallel  to  a  plane,  any  two  parallels  drawn  from 
that  line  to  the  plane  are  equal. 

21.  Parallel  Planes.     Two  planes  are  parallel  when  they 
have  no  points  in  common.     If  two  planes  are  parallel, 
either  plane  is,  by  definition,  parallel  to  all  the  lines  in 
the  other ;    also,  since  any  line  in  one  of  the  planes  is 
parallel  to  the  other  plane,  it  cannot  meet  any  line  in  that 
other  plane,  but  is  parallel  only  to  such  of  those  lines  as 
are  coplanar  with  it.     Why? 

12.  What  is  known  about  the  relative  positions  of  two  lines  that 
are  parallel  to  the  same  plane  ? 


240  LINES   AND   PLANES 

22.  Proving  a  Line  to  lie  in  a  Plane.  It  has  already 
been  said  that  a  line  is  in  a  plane  if  the  plane  contains 
two  of  its  points.  There  are  at  present  three  other  ways 
in  which  a  line  can  be  shown  to  be  in  a  plane. 

*  23.   A  line  is  in  a  plane  if  the  plane  contains  one  of  its 
points,  and  also  contains  a  second  line  to  which  the  given 
line  is  parallel. 

Compare  the  plane  determined  by  the  parallels,  and  the 
given  plane. 

A  direct  consequence  of  this  statement  is 

*  24.   A  plane  containing  one,  and  but  one,  of  two  paral- 
lels is  parallel  to  the  other. 

For  if  the  second  parallel  should  have  a  point  in  the 
plane,  what  would  follow? 

When  this  statement  is  used  to  show  that  a  line  is 
parallel  to  a  plane,  the  following  wording  is  convenient : 
If  a  line  outside  a  given  plane  is  parallel  to  a  line  in  the 
plane,  it  is  parallel  to  the  plane. 

*  25.   A  line  is  in  a  given  plane  if  the  plane  contains  one 
of  its  points,  and  if  the  line  and  the  plane  are  both  paral- 
lel to  a  second  line. 

The  parallel  lines  determine  a  plane ;  examine  its  inter- 
section with  the  given  plane. 

*  26-  A  line  is  in  a  given  plane  if  the  plane  contains  one 
of  its  points,  and  if  the  line  and  the  plane  are  both  paral- 
lel to  a  second  plane.     Use  §  25. 

27.  The  four  methods,  thus  far  considered,  of  proving 
a  line  in  a  plane,  might  be  summed  up  as  follows : 


RELATIVE   POSITIONS   OF   LINES  AND  PLANES      241 

A  line  is  in  a  given  plane  if  one  of  its  points  is  in  that 
plane,  and 

(a)  if  a  second  point  of  the  line  is  in  the  plane ; 

(6)  if  a  second  line  parallel  to  the  given  line  is  in 
the  plane; 

(<?)  if  the  line  and  the  plane  are  both  parallel  to  the 
same  line ; 

(d)  if  the  line  and  the  plane  are  both  parallel  to  the 
same  plane. 

In  each  case  two  conditions  are  necessary,  one  being 
that  a  point  of  the  given  line  shall  be  in  the  given  plane. 

13.  If  a  line  and  a  plane  are  parallel  to  the  same  line,  they  are 
parallel  to  each  other,  or  the  line  lies  in  the  plane. 

14-  If  a  line  and  a  plane  are  parallel  to  the  same  plane,  they  are 
parallel  to  each  other,  or  the  line  lies  in  the  plane. 

28.  Transversals  of  Parallels.     The  following  proposi- 
tions concerning  transversals  of  parallel  lines  and  planes 
are  direct  consequences  of  §  27. 

*  (1)  If  a  plane  cuts  one  of  two  parallel  lines,  it  cuts  the 
other  also.     For  if  not,  it  is  parallel  to  it,  and  so  must  cdh- 
tain  the  first  line. 

*  (2)  If  a  line  cuts  one  of  two  parallel  planes,  it  cuts  the 
other  also.     For  if  not,  what  follows? 

For  a  third  case,  dealing  with  three  planes,  see  §  36. 

15.  To  construct  a  line  through  a  given  point  parallel  to  a  given 
line. 

16.  To  construct  a  line  through  a  given  point  parallel  to  a  given 
plane.  J 

17.  There  can  be  an  unlimited  number  of  lines  through  a  point 
outside  a  plane  parallel  to  that  plane. 

29.  Theorem  II.     Two  intersecting  lines  parallel  to  a 
plane  determine  a  plane  parallel  to  that  plane. 


242  LINES  AND  PLANES 

For  if  the  second  plane  met  the  given  plane,  the  two 
intersecting  lines  would  both  be  parallel  to  the  line  of 
intersection  of  the  two  planes.  Why? 

30.  COR.      Through  any  point  there  can  be  passed  a 
plane  parallel  to  any  given  plane  not  containing  that 
point. 

31.  To  Construct  a  Plane.     A  plane  is  said  to  be  con- 
structed when  its  determining  elements   have  been  con- 
structed.    If  two  intersecting  lines  have  been  drawn,  a 
plane  can   be  said  to   be  passed  through  them,  that  is, 
to  be  drawn  so  as  to  entirely  contain  them,  and  the  plane 
is  then  considered  as  having  been  constructed,  and  can  be 
used  in  any  further  construction. 

It  is  obvious  that  the  passing  of  the  plane  itself  does 
not  necessarily  require  the  drawing  of  any  additional 
lines  in  the  figure.  Construction  in  solid  geometry  dif- 
fers from  that  in  plane  geometry  in  that  it  cannot  actu- 
ally accomplish  all  the  construction  in  the  plane  of  the 
drawing,  and  so  must  explain  how  to  draw  the  determin- 
ing elements  for  all  the  planes  required,  and  how  to  con- 
struct whatever  is  to  be  drawn  in  those  planes. 

The  figure  drawn  for  a  construction  represents  the 
three-dimensional  figure  as  it  should  look  to  the  observer, 
instead  of  showing  each  part  in  its  true  relation  to  the 
other  parts.  For  example,  lines  that  are  to  be  constructed 
parallel  to  each  other,  or  perpendicular  to  each  other,  can- 
not always  be  constructed  in  the  plane  of  the  drawing  so 
as  actually  to  have  those  relations.  The  student  should 
explain  what  needs  to  be  done  in  working  in  three  dimen- 
sions, and  should  draw  a  figure  to  represent  the  result  as 
nearly  as  possible.  Reread  Preliminary  Chapter,  Section 
I,  B,  VI. 


RELATIVE   POSITIONS  OF   LINES   AND   PLANES     243 

32.  CONSTRUCTION  I.     Through  a  given-  point  to  con- 
struct a  plane  parallel  to  a  given  plane. 

18.  Use  §  27  and  §  29  to  prove  Theorem  V. 

19.  Through  a  given  point  to  draw  a  line  parallel  to  one  of  two 
given  intersecting  planes  and  meeting  the  second  given  plane  at  a 
given  distance  from  the  point. 

33.  Relative  Positions  of  Three  Planes.     Three  planes 
can  intersect  in  (1)  three  lines ;   (2)  two  lines ;   (3)  one 
line ;   (4)  no   lines,  —  that   is,  be   mutually  parallel,    see 
§  37.     Why  can  they  not  intersect  in  more  than  three 
lines  ? 

34.  Three   Planes  Intersecting   in   Three    Lines.     Each 
plane  must  intersect  each  of  the  other  planes. 

Theorem  III.     If  three  planes  intersect  in  three  lines, 
the  lines  of  intersection  are  (J?)  concurrent,  or  (2}  parallel. 


Call  the  planes  A,  B,  C,  and  the  lines  of  intersection, 
according  to  the  planes  in  which  they  lie,  ab,  be,  ca. 

(1)  If  db  and  be  meet  in  point  P,  in  how  many  of  the 
planes  does  P  lie?     Is  it  therefore  in  ca?     Why?     Note 
that  this  proves  that  the  lines  of  intersection  are  concur- 
rent if  any  two  of  them-  meet. 

(2)  If  ab  and  be  do  not  meet,  they  are  parallel.     Why? 


244  LINES   AND   PLANES 

Then  can  any  two  of  the  lines  meet,  by  the  conclusion  of 
(1)?     What  follows? 

35.  Three  Planes  Intersecting  in  Two  Lines.     Two  of 

the  planes  must  be  parallel,  with  the  third  plane  inter- 
secting each  of  the  parallel  planes.     Why  ? 

Theorem  IV.  The  intersections  of  two  parallel  planes 
with  a  third  plane  are  parallel  to  each  other. 

20.   The  sects  cut  off  on  parallel  lines  by  parallel  planes  are  equal. 

36.  Theorem  V.     Through  a  given  point  not  more  than 
one  plane  can  be  passed  parallel  to  a  given  plane. 

Suppose  two  planes  through  a  point  parallel  to  a  third 
plane,  and  examine  the  intersections  of  these  three  planes 
with  another  plane  drawn  through  the  given  point,  but 
not  containing  the  line  of  intersection  of  the  two  planes 
through  that  point.  Why  should  the  plane  be  drawn  in 
this  way  ? 

This  theorem  might  be  worded  : 

Two  intersecting  planes  cannot  both  be  parallel  to  the 
same  plane,  or,  as  a  theorem  on  transversals,  A  plane 
that  intersects  one  of  two  parallel  planes  must  intersect 
the  other  also. 

It  follows  from  this  theorem  that 

37.  COR.     Planes  parallel  to  the  same  plane  are  par- 
allel to  each  other. 

38.  Three  Planes  Meeting  in  One  Line.     It  is  evident 
that  any  number  of  planes  can  be  passed  through  one  line. 
The  leaves  of  a  book  roughly  illustrate  this  possibility. 

39.  Three  Planes  Mutually  Parallel.     That  this  is  pos- 
sible wras  shown  in  §  37. 


RELATIVE   POSITIONS   OF   LINES   AND   PLANES     245 

40.  Theorem  VI.  //  two  lines  are  parallel  to  a  third 
line,  they  are  parallel  to  each  other. 

For  what  case  is  this  already  known  ?  What  is  the  new 
case  ? 


FIRST  METHOD 

Let  lines  R  and  S  be  parallel  to  line  O.  Then  there 
are  determined  planes  RQ  and  SO.  (Why?)  S  and  a 
point  P  in  R  determine  a  third  plane.  Examine  the 
intersections  of  these  three  planes,  and  so  show  that  R  is 
parallel  to  S. 

SECOND  METHOD 


Pass  a  plane  Jf  through  S  and  a  point  P  in  R  ;  then, 

(1)  M  is  parallel  to  O  (why  ?),  and  therefore  contains 
R.     But  R  cannot  meet  S,  because  if  it  did  R  and  S 
would  be"  two  intersecting  lines  parallel  to  (7. 

THIRD  METHOD 

(2)  If  M  cuts  R,  it  also  cuts  0  (why?);  also  8,  etc. 

21.  Two  planes  each  passed  through  one  of  two  parallel  lines,  are 
either  parallel  or  meet  in  a  line  parallel  to  the  given  lines. 

22.  If  two  planes  are  parallel  to  the  same  line,  they  are  parallel  to 
each  other  or  intersect  in  a  line  parallel  to  the  given  line. 

23.  As  a  door  is  opened,  show  that  all  the  positions  of  its  outer 
edge  are  parallel  to  each  other. 

24.  Describe  the  figure  of  three  parallel  lines,  without  using  the 
word  "  parallel." 


246  LINES  AND   PLANES 

25.  If  two  planes  are  parallel  to  the  same  line,  their  intersections 
with  any  plane  through  that  line  are  parallel  to  each  other. 

26.  If  a  line  is  parallel  to  a  plane,  the  intersections  of  that  plane 
with  all  the  planes  through  the  given  line  are  parallel  to  each  other. 

41.    Lines  or  Planes  Parallel  to  the  Same  Line  or  Plane. 

All  possibilities  of  lines  and  planes  that  are  parallel  to 
the  same  line  or  plane  have  now  been  considered,  with 
the  following  results  : 

(1)  Parallel  to  the  Same  Line. 

(a)  Two  lines ;  they  are  parallel  to  each  other. 

(6)  A  line  and  a  plane ;  they  are  parallel  to  each 
other,  or  the  line  lies  in  the  plane. 

(<?)  Two  planes ;  it  is  not  known  whether  they 
intersect  or  not.  If  they  do  meet,  their  line 
of  intersection  is  parallel  to  the  given  lihe. 

(2)  Parallel  to  the  Same  Plane. 

(a)  Two  planes ;  they  are  parallel  to  each  other. 
(6)   A  line  and  a  plane ;  they  are  parallel  to  each 

other,  or  the  line  lies  in  the  plane. 
(<?)   Two  lines ;  it  is  not  known  whether  they  are 
intersecting,    parallel,   or   non-coplanar.     If 
they   meet,  their   plane   is   parallel   to   the 
given  plane. 
To  sum  up:     If  the  three  are  alike,  —  that  is,  all 

lines  or  all  planes,  —  they  are  all  parallel. 
If  two  unlike  are  parallel  to  a  third,  the  two  are 
parallel,  or  the  line  lies  in  the  plane.  It  appears 
that  a  line  being  parallel  to  a  plane,  and  a  line 
being  in  the  plane,  are  usually  special  cases  of 
one  general  proposition. 

If  two  alike  are  parallel  to  a  third  unlike,  the  rela- 
tive position  of  the  two  is  not  fixed. 


RELATIVE    POSITIONS   OF   LINES   AND   PLANES      247 

42.  Theorem  VII.     If  two  intersecting   lines   in   one 
plane  are  respectively  parallel  to  two  intersecting  lines 
in  a  second  plane, 

(1)  the  two  planes  are  parallel ; 

(2)  the  angles  in  one  plane  are  respectively  equal  to 
or  supplemental  to  the  angles  in  the  other  plane  (accord- 
ing to  their  relative  positions  with  regard  to  the  vertices). 

For  (1),  see  Theorem  II. 

(2)  What  method  of  proving  angles  equal  can  be  ap- 
plied even  if  the  angles  are  in  different  planes?  Cut  off 
equal  lengths  on  the  arms  of  a  correspondingly  placed 
pair  of  angles,  find  what  planes  are  determined,  and  see 
whether  the  material  necessary  for  this  method  can  be 

obtained. 

• 

43.  The  Possibility  of  passing  Planes  through  One  Line 
Parallel  to  a  Second  Line. 

(1)  If  the  lines  intersect,  no  plane  can  be  drawn  through 
one  line  parallel  to  the  other.     Why  ? 

(2)  If  the  lines  are  parallel,  any  plane  through  one  of 
the  lines  (except  the  plane  determined  by  the  two  lines) 
is  parallel  to  the  other  line.     Why? 

(3)  If  the  lines  are  non-coplanar,  then  Theorem  VIII 
follows. 

44.  Theorem  VIII.     If  two  lines  are  non-coplanar,  one 
and  but  one  plane  can  be  passed  through  either  parallel 
to  the  other. 

From  any  point  in  one,  draw  a  line  parallel  to  the 
other.  How  can  this  be  done,  why  is  it  done,  and  what 
conclusion  follows  ? 

Also,  this  is  the  only  plane  possible,  for  a  plane  through 
the  first  given  line  parallel  to  the  other  given  line  must 
contain  the  line  constructed*  Why  ? 


248  LINKS   AND   PLANES 

45.  CONST.  II.     Through   one  of  two  given    non-co- 
planar  lines  to  draw  a  plane  parallel  to  the  other. 

46.  Theorem  IX.     Through  a  given  point,  one  and  but 
one  plane  can  be  passed  parallel  to  two  non-coplanar  lines. 

Follow  the  same  method  of  proof  as  in  Theorem  VIII. 

In  what  special  case  will  the  plane  contain  one  of  the 
lines  instead  of  being  parallel  to  it  ?  Note  the  similarity 
to  the  conclusion  in  §  41,  1,  (5)  and  2,  (5). 

47.  CONST.  III.     Through  a  given  point  to  construct 
a  plane  parallel  to  two  given  non-coplanar  lines. 

48.  Theorem  X.     //  two  lines  are  cut  by  three  parallel 
planes,  their  corresponding  sects  are  proportional. 

If  the  lines  are  parallel  or  concurrent,  this  has  been 
proved  in  plane  geometry.  Explain  why.  What  is  the 
third  case  ?  In  order  to  determine  planes  (why  are  they 
needed  ?),  draw  a  third  line  so  that  it  will  determine  a 
plane  with  each  of  the  given  lines,  then  use  plane  geom- 
etry. Find  two  ways  to  draw  this  auxiliary  line. 

49.  COR.    If  lines  from  a  point  (sometimes  called  a 
sheaf  of  lines)  are  cut  by  parallel  planes,  the  correspond- 
ing sects  on  any  two  lines  are  proportional. 

27.  The  planes  of  a  biplane  are  six  feet  apart,  and  two  twelve- 
foot  braces  pass  through  the  planes  and  to  a  point  four  feet  below  the 
lower  plane.     What  length  is  cut  off  on  them  by  the  planes? 

28.  If  two  congruent  parallelograms  lie  in  parallel  planes  with  their 
sides  respectively  parallel,  how  many  planes  are  determined  by  using 
their  sides  in  all  possible  combinations  ? 

29.  How  many  intersections  have  the  planes  determined  in  Ex.  28, 
and  what  can  be  told  of  these  intersections?     Do  any  of  these  inter- 
sections determine  other  planes?     Examine  the  intersections  of  these 
planes  with  each  other  and  with  the  original  planes. 


SECTION    III.     LINES    PERPENDICULAR   TO    PLANES 

(At  the  discretion  of  the  teacher  §§  50  and  51  may  be  omitted,  §  53 
being  assumed.) 

*  50.  The  Shortest  Line  from  a  Point  to  a  Plane.  From 
any  given  external  point  to  a  plane,  tliere  is  one  sect 
shorter  than  any  other. 

P/ 


The  sects  from  p  to  plane  M  are  of  various  lengths,  for 
if  a  line  is  drawn  in  M  the  sects  from  P  to  points  on  that 
line  are  of  various  lengths.  (Why  ?)  Therefore  there  is 
some  one  length  that  is  the  shortest  possible  length  from 
P  to  M.  + 

No  two  sects,  as  PA  and  P.B,  could  both  be  of  this 
shortest  length,  for  they  determine  a  plane  meeting  M  in 
AB,  and  being  equal,  both  are  longer  than  the  perpendic- 
ular from  P  to  AB.  (Explain  wiry.) 

Therefore  but  one  sect  has  the  shortest  possible  length 
from  P  to  Jlf,  that  is,  there  is  one  sect  from  P  to  M  that  is 
shorter  than  any  other. 

*  51.  The  shortest  sect  from  a  given  external  point  to 
a  given  plane  is  perpendicular  to  all  lines  in  the  plane 
through  its  foot. 

249 


250  LINES  AND    PLANES 

For  if  there  were  a  line  through  its  foot  to  which  it  was 
not  perpendicular,  the  perpendicular  from  the  given  point 
to  that  line  would  be  shorter,  which  is  not  possible. 

52.  Line  Perpendicular  to  a  Plane.     A  line  is  perpen- 
dicular to  a  plane  (and  the  plane  is  perpendicular  to  the 
line)  when  the  line  meets  the  plane  and  is  perpendicular 
to  all  lines  in  the  plane  through  its  foot.     When  a  line 
meets  a  plane  and  is  not  perpendicular  to  it,  it  is  oblique 
to  the  plane.     The  term  normal  is  sometimes  used  instead 
of  perpendicular. 

53.  Existence  of  a  Line  Perpendicular  to  a  Plane,  and  of 
a  Plane  Perpendicular  to  a  Line.      In  §§  50  and  51,  it  was 
proved  that 

From  any  given  external  point  to  any  given  plane, 
there  is  a  line  perpendicular  to  that  plane. 

It  also  follows  that 

At  any  given  point  of  a  given  plane,  there  is  a  line 
perpendicular  to  that  plane,  and 

At  any  given  point  in  a  given  line,  there  is  a  plane 
perpendicular  to  that  line. 

For,  since  there  can  be  a  line  perpendicular  to  a  plane, 
that  figure  can  be  superposed  so  as  to  make  the  line  per- 
pendicular to  any  plane  at  any  point  in  it,  or  so  as  to  make 
the  plane  perpendicular  to  any  line  at  any  point  in  it. 
Explain  how  this  can  be  done. 

To  sum  up,  it  is  possible  to  assume  in  a  figure: 

(1)  A  line  perpendicular  to  any  plane 
(a)  from  any  external  point  •, 

(5)   at  any  point  in  the  plane. 

(2)  A  plane.perpendicular  to  any  line 
(a)  at  any  point  in  the  line. 


LINES  PERPENDICULAR   TO   PLANES  251 

(5)   It  has   not   yet  been  shown,  but  will  follow 
from  §§62  and  63,  that  there  can  be  a  plane 
.perpendicular  to  any  line  from  any  external 
point. 

54.  Theorem  XI.     Two  intersecting  lines  Cannot  be  per- 
pendicular to  the  same  plane. 

The  point  of  intersection  can  be  either  in  or  outside 
the  plane.  Suppose  two  perpendiculars  to  the  given  plane 
through  this  point ;  they  determine  a  plane,  and  have 
what  relation  to  the  intersection  of  the  two  planes  ? 

55.  Theorem  XII.      Two     intersecting   planes    cannot 
both  be  perpendicular  to  the  same  line. 

Pass  a  plane  through  the  given  line  so  as  to  meet  the 
two  intersecting  planes  in  intersecting  lines.  Can  these 
lines  both  be  perpendicular  to  the  given  line? 

It  follows  that 

56.  COR.     Planes  perpendicular  to  the  same  line  are 
parallel. 

57.  Theorem  XIII.     A  line  perpendicular  to  one  of  two 
parallel  planes  is  perpendicular  to  the  other  also. 

For  it  meets  the  second  plane  (why?),  and  by  the  use 
of  parallels  it  can  be  shown  to  be  perpendicular  to  any 
line  in  that  plane.  Or, 

Draw  a  plane  perpendicular  to  the  line  at  the  second 
intersection  and  prove  that  it  coincides  with  the  given 
plane. 

80.   Planes  perpendicular  to  intersecting  lines  are  not  parallel. 

58.  Theorem  XIV.     If  one  of  two  parallel  lines  is  per- 
pendicular to  a  plane,  the  other  also  is  perpendicular  to 
that  plane. 


252  LINES   AND  PLANES 

It  meets  the  plane  (why?);  then  use  parallels  to  prove 
it  perpendicular  to  every  line  through  its  foot. 

81.  If  an  oblique  line  cuts  two  parallel  planes,  the  angles  it  makes 
with  the  lines  through  its  foot  in  one  plane  are  equal  to  the  corre- 
sponding angles  it  makes  with  the  lines  through  its  foot  in  the  other 
plane.     What  determines  which  are  the  corresponding  angles  in  the 
two  planes? 

59.  Theorem  XV.     Lines  perpendicular  to  the  same 
plane  are  parallel. 

Take  any  two,  and  through  the  foot  of  one  draw  a  par- 
allel to  the  other.  What  follows  ? 

82.  All  perpendiculars  from  a  given  line  to  a  given  plane  are  co- 
planar.     In  what  positions  might  the  given  line  lie? 

33.   Sects  of  perpendiculars  between  parallel  planes  are  equal. 

60.  Theorem  XVI.     Any  line  perpendicular  to  a  given 
line  at  a  given  point  lies  in  the  plane  perpendicular  to 
the  line  at  that  point. 

Pass  a  plane  through  the  two  lines,  and  examine  the 
intersections  of  the  two  planes. 

Note  that  this  is  a  fifth  method  of  showing  that  a  line 
is  in  a  plane.  It  might  be  stated  as  follows : 

A  line  is  in  a  plane  if  it  has  one  point  in  the  plane, 
and  if  the  line  and  the  plane  are  both  perpendicular  to 
the  same  line  at  the  same  point.  Show  that  if  the  last 
four  words  are  omitted,  the  statement  is  still  true. 

61.  COR.  1.    All    the    perpendiculars    that    can    be 
drawn  to  a  given  line  at  a  given  point  are  coplanar. 

62.  COR.  2.     A  line  perpendicular  to  two  intersecting 
lines  is  perpendicular  to  their  plane.     For  they  lie  in  and 
determine  the  perpendicular  plane. 


LINES   PERPENDICULAR   TO   PLANES  253 

34.  Two  lines  perpendicular  to  the  same  line  at  the  same  point  de- 
termine a  plane  perpendicular  to  the  line  at  that  point. 

35.  One  arm  of  a  right  angle  revolved  about  the  other  arm  as  an 
axis  generates  a  plane.     Does  the  arm  of  any  other  angle  generate  a 
plane? 

36.  The  hand  of  a  clock  revolves  in  a  plane.     Does  the  arm  of  a 
hoisting  crane  revolve  in  a  plane? 

63.    CONST.   IV.     To  draw  a  plane  through  a  given 
point  perpendicular  to  a  given  line. 

In  how  many  positions  might  the  point  be  ?    How  much 
is  needed  to  determine  the  required  plane  ? 


SECTION   IV.     ANGLES   BETWEEN   PLANES 


64.  Dihedral  Angles.  An  angle  between  two  intersect- 
ing planes  is  called  a  dihedral  angle.  The  planes  are 
called  the  faces,  and  their  intersection  is  called  the  edge  of 
the  dihedral  angle. 


Two  intersecting  planes  evidently  form  four  dihedral 
angles,  just  as  two  intersecting  lines  form  four  plane 
angles.  In  the  figure,  the  planes  M  and  N,  intersecting 
in  XY,  form  the  dihedral  angles  indicated  by  the  arrows 
as  A,  B,  O,  and  D.  A  dihedral  angle  may  be  denoted  by 
naming  its  faces,  with  or  without  the  use  of  the  intersec- 
tion, as  dihedral  angle  MN,  or  dihedral  angle  M-XY-N. 
When,  as  in  this  figure,  there  might  arise  confusion  as  to 
which  angle  was  intended,  a  second  letter  may  be  used  for 
each  plane,  as  Mf  and  Nf  in  the  figure,  and  their  order  in 
the  naming  of  the  angle  will  determine  which  angle  is 
meant.  Thus,  angle  (7  could  be  named  M'-XY-N*,  the 
direction  of  rotation  being  considered,  as  in  plane  geom- 
etry, counterclockwise  (that  is,  in  the  direction  opposite 
to  that  in  which  the  hands  of  a  clock  rotate). 

254 


ANGLES   BETWEEN    PLANES  255 

As  the  size  of  a  plane  angle  does  not  depend  on  the 
length  of  its  arms,  so  the  size  of  a  dihedral  angle  does  not 
depend  011  the  extent  of  its  faces,  but  only  on  the  amount 
of  rotation  necessary  for  a  plane  to  rotate  about  the  inter- 
section from  one  face  to  the  other. 

Familiar  illustrations  of  dihedral  angles  are,  the  angle 
between  two  leaves  of  an  open  book,  the  angle  between 
two  walls  of  a  room,  the  angle  between  two  planes  of  a 
roof,  the  various  angles  made  as  a  desk  cover  is  raised  or 
as  a  door  is  opened,  etc. 

65.  Plane,  or  Measuring,  Angle.     If  at  a  point  in  the 
edge  of  a  dihedral  angle,  two  perpendiculars,  one  in  each 
face,  are  drawn  to  the  edge,  the  angle  between  the  perpen- 
diculars is  called  the  plane  angle,  or  the  measuring  angle  of 
the  dihedral  angle.     This  plane  angle  evidently  lies  in, 
and  its  arms  determine,  a  plane  perpendicular  to  the  edge. 

It  is  evident  that  all  plane  angles  of  the  same  dihedral 
angle  are  equal.  Why? 

66.  Theorem  XVII.     Plane  angles  of  equal  dihedral 
angles  are  equal. 

Superpose. 

37.  The  greater  of  two  unequal  dihedral  angles  has  the  greater 
plane  angle. 

67.  Theorem  XVIII.     Two  dihedral  angles  are  equal 
if  their  plane  angles  are  equal. 

38.  Vertical  dihedral  angles  are  equal.     (See  def .  in  §  70.) 

68.  Theorem  XIX.     Two  dihedral  angles  are  propor- 
tional to  their  plane  angles. 

Assume  a  common  divisor  for  the  commensurable  case. 


256  LINES  AND  PLANES 

69.  Measurement  of  a  Dihedral  Angle.    Since  the  amount 
of  rotation  in  a  dihedral  angle  is  the  same  part  of  a  com- 
plete rotation  about  its  edge  that  its  plane  angle  is  of  a 
perigon  (Th.  XIX),  the  plane  angle  is  the  measure  of  the 
dihedral  angle,  this  complete  rotation  being  considered  as 
a  standard  by  which  to  measure.     It  is  on  this  account  that 
the  plane  angle  is  called  the  measuring  angle  of  the  dihedral 
angle. 

It  is  evident,  therefore,  that  propositions  concerning 
dihedral  angles  will  usually  be  proved  by  the  use  of  their 
measuring  angles. 

70.  Terms  used  for  Plane  Angles  that  apply  also  to  Di- 
hedral Angles.     On  account  of  the  dependence  of  dihedral 
angles  on  their  plane   angles,  a  dihedral  angle  is  called 
acute,  right,  obtuse,  straight,  or  reflex,  according  as  its  plane 
angle  is  acute,  right,  obtuse,  straight,  or  reflex;  also  two 
dihedral  angles  are  adjacent,  vertical,  complementary,  sup- 
plementary, or  explementary,  according  as  their  measuring 
angles   fulfill    the    conditions  of  the  definitions  of  these 
terms. 

If  the  dihedral  angle  between  two  planes  is  a  right  di- 
hedral angle,  the  planes  are  said  to  be  perpendicular  to 
each  other. 

71.  Theorem   XX.     If  a   line   is   perpendicular  to  a 
plane,  any  plane  through  that  line  is  perpendicular  to 
the  given  plane. 

How  can  planes  be  proved  perpendicular?  What  aux- 
iliary line  does  this  necessitate  ? 

Note  that  this  is  a  second  method  of  proving  planes 
perpendicular,  and  that  it  is  simpler  to  apply  than  the 
definition,  since  that  requires  the  use  of  two  lines,  while 
this  method  requires  but  one. 


ANGLES   BETWEEN   PLANES  257 

89.  How  many  intersecting  planes  can  be  perpendicular  to  the 
same  plane? 

72.  Theorem   XXI.     A  line  drawn  in  one  of  two  per- 
pendicular planes,  perpendicular   to  their  intersection, 
is  perpendicular  to  the  other  plane. 

This  proves  that  each  of  two  perpendicular  planes  con- 
tains an  unlimited  number  of  lines  perpendicular  to  the 
other  plane.  Are  all  the  lines  in  one  plane  perpendicular 
to  the  other?  Are  all  the  lines  that  meet  the  intersection 
perpendicular  to  the  other  plane  ? 

40.  If  a  line  and  a  plane  intersect,  a  plane  perpendicular  to  one 
is  not  perpendicular  to  the  other. 

73.  COR.  1.     If  two  planes  are  perpendicular,  a  line 
perpendicular  to  one  of  them  through  a  point   in  the 
other,  lies  wholly  in  that  other. 

Use  the  theorem. 

Stated  like  the  other  methods  of  proving  a  line  in  a 
plane,  this  becomes :  A  line  is  in  a  plane  if  it  has  one 
point  in  that  plane,  and  if  the  line  and  the  plane  are 
both  perpendicular  to  the  same  plane. 

74.  COR.   2.     If  two  intersecting  planes  are  each  per- 
pendicular to  a  third  plane,  their  intersection   is   per- 
pendicular to  that  plane. 

What  is  known  of  a  perpendicular  to  the  third  plane 
through  any  point  of  the  given  intersection? 

41.  Can  planes  perpendicular  to  intersecting  planes  be  parallel? 
Explain. 

75.  COR.    3.     If  two  intersecting  lines  are  perpendic- 
ular to  the  faces  of  a  dihedral  angle,  they  determine  a 
plane  t^t  contains  one  of  the  measuring  angles  of  that 
dihedral  angle. 


258  LINES   AND   PLANES 

76.  Theorem   XXII.     Through    any   line  not    perpen- 
dicular to  a  given  plane,  one,  and  but  one,  plane  can  be 
passed  perpendicular  to  that  plane. 

Use  the  simplest  method  of  determining  a  plane  per- 
pendicular to  the  given  plane,  then  show  that  a  perpen- 
dicular plane  must  contain  these  determining  lines. 

77.  Projections.     The  projection  of  a  point  upon  a  plane 
is  the  foot  of  the  perpendicular  from  that  point  to  the 
plane.     The  projection  of  a  line  upon  a  plane  is  the  locus 
of  the  projections  of  all  its  points  upon  that  plane. 

78.  Theorem   XXIII.     The  projection  upon  a  plane  of 
a  line  that   is  not    perpendicular  to  that  plane   is   a 
straight  line. 

Prove  (1)  the  projections  of  all  its  points  will  lie  in 
one  line  ;  (2)  any  point  in  that  line  is  the  projection  of 
some  point  of  the  given  line.  What  would  be  the  pro- 
jection of  a  perpendicular  upon  a  plane  ? 

NOTE.  The  following  group  of  exercises  on  projections  is  closely 
associated  with  the  elements  of  descriptive  geometry. 

Griven  two  planes  perpendicular  to  each  other  : 

42.  Find  the  line  of  which  two  given  lines,  one  in  each  plane  (but 
not  both  perpendicular  to  the  intersection)  are  the  projections  upon 
those  planes. 

43.  If  a  line  is  perpendicular  to  one  of  the  planes,  its  projection 
upon  the  other  is  perpendicular  to  the  intersection. 

44-  If  ft  h'ne  is  parallel  to  one  of  the  planes,  its  projection  upon 
the  other  plane  is  parallel  to  the  intersection. 

45.  If  a  line  is  parallel  to  both  planes,  its  projections  upon  them 
are  parallel  to  each  other. 

46.  Tf  two  intersecting  lines  are  projected  upon  both  planes,  the 
line  joining  the  intersection  of  their  projections  upon  one  plane  with 


ANGLES  BETWEEN    PLANES  259 

the  intersection  of    their  projections  upon  the  other  plane  is   in  a 
plane  perpendicular  to  the  intersection  of  the  given  planes. 

47.  If  a  third  plane  is  perpendicular  to  either  of  the  given  planes, 
its  intersection  with  the  other  plane  is  perpendicular  to  the  intersec- 
tion of  the  given  planes. 

48.  If  a  given  line  is  perpendicular  to  a  third  plane,  its  projections 
upon  the  given  planes  are  respectively  perpendicular  to  the  intersec- 
tions of  the  third  plane  with  the  given  planes. 

49.  The  projection  of  any  line  in  one  upon  the  other  is  the  inter- 
section. 

79.  Theorem    XXIV.       Of  all  lines  drawn    from  a 
given  external  point  to  a  given  plane, 

(a)  The  perpendicular  is  the  shortest. 

(b)  Those  making  equal  angles  with  the  perpendicu- 
lar, or  having  equal  projections  on  the  plane,  are  equal. 

(c)  Those  making  unequal  angles  with   the  perpen- 
dicular, or  having  unequal  projections  on  the  plane,  are 
unequal,  the  one  makmg  the  greater  angle,  or  having 
the  longer  projection,  being  the  longer. 

Use  plane  geometry. 

80.  Distance  from  a  Point  to  a  Plane.     The  length  of 
the  perpendicular  from  a  point  to  a  plane  is  called  the  dis- 
tance from  the  point  to  the  plane.     Why  ? 

81.  COR.     Equal   obliques  from   a  point  to  a  plane 
make  equal  angles  with  the  perpendicular,  and  have 
equal  projections  upon  the  plane. 

Unequal  obliques  from  a  point  to  a  plane  make  un- 
equal angles  with  the  perpendicular,  and  have  un- 
equal projections  upon  the  plane,  the  longer  oblique 
making  the  greater  angle  and  having  the  longer  pro- 
jection. 

What  relation  has  this  to  (6)  and  (<?)  of  the  theorem  ? 


260  LINES  AND  PLANES 

50.  All  points  in  a  given  plane  at  a  fixed  distance  from  a  given 
external  point  lie  in  a  circle.  All  points  in  a  circumference  are  equi- 
distant from  any  point  on  the  perpendicular  to  their  plane  at  the 
center  of  the  circle. 

51.'  From  any  point  not  in  the  perpendicular  at  the  center  of  a 
circle,  only  two  points  in  the  circumference  are  equidistant.  How 
could  the  second  of  such  a  pair  of  points  be  found  if  the  first  is 
given  ? 

82.  Theorem   XXV.     The  acute  angle  that  an  oblique 
makes  with  its  projection  upon  a  plane  is  the  least  angle 
it  makes  with  any  line  in  the  plane. 

How  can  angles  not  in  the  same  triangle  be  compared 
in  size  ?  Use  a  perpendicular  from  the  given  line  to  its 
projection. 

83.  The  Angle  between  a  Line  and  a  Plane.     The  lesser 
of  the  two  angles  that  a  line  makes  with  its  projection 
upon  a  plane  is  called  its  angle  with  the  plane,  or  its  incli- 
nation to  the  plane.     Why  is  this  angle  used  ? 

52.  Which  is  the  greatest  angle  that  an  oblique  line  makes  with 
any  line  in  a  given  plane? 

53.  Parallel  lines  are  equally  inclined  to  a  plane. 

54.  A  line  is  equally  inclined  to  two  parallel  planes. 

55.  If  one  of  two  parallels  meets  a  plane,  and  the  other  meets  a 
parallel  plane,  their  inclinations  to  those  planes  are  equal. 

56.  Equal  obliques  from  a  point  to  a  plane  make  equal  angles 
with  that  plane. 

84.  Theorem  XXVI.      If  a  line  meets   its   projection 
upon  a  plane,  the  line  in  the  plane  perpendicular  to  one 
of  them  at  their  intersection  is  perpendicular  to  the 
other  also. 

The  line  and  its  projection  determine  a  plane.  Prove 
the  two  planes  perpendicular  to  each  other.  Or, 


ANGLES   BETWEEN  PLANES  261 

Cut  off  equal  sects  on  the  perpendicular,  and  then  use 
obliques.  This  proof  is  practically  the  same,  although  in 
reversed  order,  in  the  two  cases. 

57.  A  line  oblique  to  a  plane  is  concurrent  with  two  lines  in  the 
plane,  and  its  projection  bisects  the  angle  between  them.     Prove  that 
it  makes  equal  angles  with  those  lines. 

58.  Any  line   in    one  of   two  perpendicular  planes,  meeting  the 
intersection,  is  perpendicular  to  an  arm  of  a  measuring  angle. 

59.  If  from  a  point  outside  a  plane,  perpendiculars  are  drawn  to 
the  arms  of  a  right  angle  in  that  plane,  they  include  an  acute  angle. 

60.  If  from  a  point  outside  a  plane,  perpendiculars  are  drawn  to 
the  arms  of  an  angle  in  that  plane,  they  include  an  angle  less  than  its 
supplement. 

61.  If  a  figure  has  four  straight  line  sides  and  four  right  angles, 
it  lies  in  one  plane,  and  is  a  rectangle. 

62.  If  from  a  point  in  one  of  two  oblique  planes  a  perpendicular  is 
drawn  to  the  intersection,  and  a  second  perpendicular  is  drawn  to  the 
other  plane,  the  plane  determined  by  those  lines  is  perpendicular  to 
both  the  given  planes  and  to  their  intersection. 

85.  CONST.  V.     To  draiv  a  plane  through  a  given 
point  perpendicular  to  a  given  plane. 

Draw  any  line  in  the  plane,  then  draw  a  plane  through 
the  given  point  perpendicular  to  that  line.  Why  are  the 
planes  perpendicular? 

86.  CONST.  VI.     To  draw  a  line  through   a  given 
point  perpendicular  to  a  given  plane. 

First  draw  a  plane  through  the  point  perpendicular  to 
the  given  plane.  How  can  a  line  be  drawn  in  this  plane 
so  as  to  be  perpendicular  to  the  given  plane  ? 

How  can  this  be  used  to  construct  a  plane  through  a 
given  line  perpendicular  to  a  given  plane  ? 

63.  To  construct  a  plane   through  a  given  point  so  as  to  make 
equal  angles  with  two  given  intersecting  planes. 


262  LINES  AND  PLANES 

87.  Theorem  XXVII.  Two  non-coplanar  lines  have 
one,  and  but  one,  common  perpendicular,  and  it  is  the 
shortest  line  that  can  be  drawn  between  them. 

Pass  a  plane  through  each  of  the  lines  parallel  to  the 
other,  and  a  plane  through  each  of  the  lines  perpendicular 
to  these  parallel  planes.  Where  is  the  common  perpen- 
dicular? Show  that  it  is  the  only  one  possible.  The 
pupil  may  be  able  to  prove  this  theorem  equally  well  with 
but  two  of  these  planes  drawn. 

The  length  of  their  common  perpendicular  is  called  the 
distance  between  two  non-coplanar  lines.  Why  ? 

64-  If  a  line  is  parallel  to  a  plane,  its  distances  from  all  the  lines 
in  the  plane  that  are  non-coplanar  with  it  are  equal. 


SECTION  V.     LOCUS   OF   POINTS 

88.  Loci.     The  proof   of  a  locus  theorem  must,  as  in 
plane  geometry,  consist  of  two  parts,  —  the  proof  that 

(a)  Every  point  that  fulfills  the  given  conditions  lies 
on  a  certain  figure  (usually  consisting  of  one  or  more 
points,  lines,  or  surfaces) ;  and 

(6)  Every  point  on  that  figure  fulfills  the  conditions. 

The  negative  converse  of  either  of  these  statements 
can  be  proved  in  its  stead  if  that  seems  desirable,  but 
such  a  possibility  is  not  likely  to  arise. 

One  locus  theorem,  Theorem  XXIII,  has  already  been 
demonstrated,  but  as  it  is  used  to  show  that  a  projection 
is  straight,  rather  than  to  find  a  locus,  it  is  not  classed 
with  the  theorems  of  this  section. 

65.  Find  the  locus  of  points  at  a  fixed  distance  from  a  given  plane. 

66.  Find  the  locus  of  points  at  fixed  distances  from  each  of  two 
given  planes. 

89.  Theorem   XXVIII.     The   locus  of  points   equally 
distant  from  two  points  is  the  plane  perpendicular  to 
the  line  joining  them,  at  its  middle  point. 

Use  plane  geometry,  with  what  is  known  about  a  plane 
that  is  perpendicular  to  a  line,  to  prove  that 

(a)  Any  point  equidistant  from  the  two  points  is  in 
that  plane. 

263 


264  LINES     AND   PLANES 

(b)  Any  point  in  that  plane  is  equidistant  from  the 
two  points. 

Note  that  this  proves  also  that 

(<?)  Any  point  not  in  the  plane  is  not  equidistant  from 
the  two  points. 

(d)  Any  point  not  equidistant  from  the  two  points  is 
not  in  the  plane. 

90.  Theorem   XXIX.     The  locus  of  points  equidistant 
from  three  non-collinear  points  is  the  perpendicular  to 
their  plane  at  the  circumcenter  of  the  triangle  formed. 

Prove  it  by  Theorem  XXVIII,  applied  twice,  or  use 
projections. 

91.  Theorem  XXX.      The  locus  of  points  equidistant 
from  four  non-coplanar  points,  no  three  of  which  are  col- 
linear,  is  a  point. 


FIRST  METHOD 

Let  the  points  be  A^  #,  C,  and  D.  Erect  the  perpen- 
dicular OR  to  plane  ABC  at  the  circumcenter  o,  and  draw 
plane  IT,  the  perpendicular  bisector  of  AD.  Since  AD 
intersects  plane  ABC  (why?),  K  is  not  perpendicular  to 
ABC  (why?)  and  meets  OR  at  some  point  P.  What  is 
known  of  P? 


LOCUS   OF   POINTS  265 


SECOND  METHOD 

Draw  AB,  AC,  and  A D,  and  their  perpendicular  .bisect- 
ing planes  i,  3f,  and  N.  Then  no  two  of  these  planes 
are  parallel.  (Why?)  They  intersect  either  (a)  in  one 
line,  (5)  in  three  parallel  lines,  or  (c)  in  three  concur- 
rent lines,  and,  therefore,  all  in  a  point.  But  the  inter- 
section of  L  and  M  would  be  perpendicular  to  the  plane 
ABC,  and  the  intersection  of  M  and  N  would  be  perpen- 
dicular to  the  plane  A  CD,  therefore  (a)  and  (#)  are  im- 
possible, and  (c)  is  the  only  possibility. 

THIRD  METHOD 

Draw  triangles  ABC  and  BCD ;  erect  perpendiculars  to 
their  planes  at  the  circumcenters  of  those  triangles,  and 
prove  that  these  perpendiculars  lie  in  a  plane  and  inter- 
sect. This  is  probably  the  hardest  method  of  the  three, 
but  all  are  good  practice  in  loci.' 

Note  that  if  the  six  sects  joining  the  four  points  are 
drawn,  each  plane  that  is  the  perpendicular  bisector  of 
one  of  these  sects  must  contain  the  point  equidistant 
from  all  the  points;  therefore  these  six  planes  have  inter- 
sections that  are  concurrent  at  this  point. 

67.  Find  the  points  in  a  given  plane  that  are  equidistant  from  two 
given  points. 

68.  Find  the  locus  of  points  in  a  given  plane,  equidistant  from 
three  points  not  in  a  line. 

69.  Find  the  locus  of  points  equidistant  from  two  lines  (a)  parallel, 
(&)  intersecting.     Remember  that  distance  from  a  point  to  a  line 
always  means  distance  along  a  perpendicular. 

70.  Find    the   locus    of    points    equidistant    from    three    concur- 
rent lines. 


266 


LINES   AND   PLANES 


92.  Theorem  XXXI.  The  locus  of  points  equidistant 
from  two  intersecting  planes  is  the  pair  of  planes  that 
bisect  their  dihedral  angles. 


(1)  To  prove.     A  point  equidistant  from  two  intersect- 
ing planes  lies  in  a  plane  bisecting  the  dihedral  angle 
between  whose  faces  the  point  lies. 

Let  the  planes  M  and  N  meet  in  XT,  and  let  point  P  be 
equidistant  from  M  and  jv,  that  is,  let  the  perpendiculars 
PA  and  PC  to  those  planes  be  equal.  Plane  B  is  deter- 
mined by  P  and  XY\  it  is  then  necessary  to  prove  that 
B  bisects  the  dihedral  angle  MN.  PA  and  PC  determine 
the  plane  K  of  the  measuring  angle  AEG.  (Why?)  Is 
P  equidistant  from  the  arms  of  the  measuring  angle 
ARC?  (Why?)  Where  then  must  P  lie ? 

(2)  Prove   the    converse,    using   the    same  method  of 
construction. 

71.  Find  the  locus  of  points  equidistant  from  two  parallel  planes. 

72.  Where  do  the  midpoints  of  all  transversals  of  two  non-coplanar 
lines  lie?     Is  this  therefore  a  locus? 

NOTE.  The  locus  of  points  equidistant  from  two  points,  from  two 
coplanar  lines,  and  from  two  planes  has  been  found.  It  is  not  pos- 
sible, however,  in  elementary  geometry,  to  find  the  locus  of  points 
equidistant  from  two  different  elements,  as  a  point  and  a  line,  a  line 
and  a  plane,  a  point  and  a  plane,  or  from  two  non-coplanar  lines.  Such 
loci  are  discussed  in  analytical  geometry  and  related  subjects. 


SUMMARY   OF   PROPOSITIONS  267 

73.  If  three  planes  meet  in  one  point,  show  that  the  bisectors  of 
the  dihedral  angles  formed  are  concurrent  in  a  line. 

74.  If  four  planes  meet,  each  three  in  a  point,  show  that  the 
bisectors  of  the  six  dihedral  angles  meet  in  a  point. 

93.  COR.   1.     To  find  the  locus  of  points  equidistant 
from  three  planes  that  meet  in  one  point. 

94.  COR.   2.      To  find  the  locus  of  points  equidistant 
from  four  planes,  each  three  of  which  meet  in  one  point. 

75.  If  three  planes  intersect  in  three  concurrent  lines,  show  that 
the  planes  perpendicular  to  the  given  planes  through  the  bisectors  of 
the  plane  angles  between  their  lines  of  intersection  meet  in  a  line. 

76.  Find  a  point  in  one  of  two  given  non-coplanar  lines  equidis- 
tant from  two  given  points  in  the  other. 

77.  Find  the  locus  of  points  in  a  given  plane  equidistant  from  two 
given  coplanar  lines  outside  the  plane. 

78.  Find  the  locus  of  a  point  such  that  the  sum  of  the  squares  of 
its  distances  from  two  given  fixed  points  shall  be  constant. 

79.  Find  the  locus  of  a  point  such  that  the  difference  of  the  squares 
of  its  distances  from  two  given  fixed  points  shall  be  constant. 

80.  Find  the  locus  of  points  equidistant  from  two  given  planes  and 
from  two  given  points;  from  two  given  coplanar  lines  and  from  two 
given  points ;  from  two  given  coplanar  lines  and  from  two  planes. 

81.  Find  the  locus  of  points  at  a  fixed  distance  from  a  given  plane 
and  equidistant  from  two  given  points;    from  two  given  coplanar 
lines;  from  two  given  planes. 

95.  SUMMARY  OF  PROPOSITIONS 

(Numbers  in  parentheses  refer  to  black-faced  section  numbers.) 
I.   THE  STRAIGHT  LINE  : 
(1)  Lines  parallel : 

A  line  in  one  of  two  intersecting  planes  and  parallel  to  the 

other  is  parallel  to  the  intersection  (20). 
Two   lines   parallel   to   the    same  line  are  parallel  to  each 
other  (40). 


268  LINES  AND   PLANES 

Two  lines  perpendicular  to  the  same  plane  are  parallel  to 
each  other  (59). 

The  intersections  of  two  parallel  planes  with  a  third  plane 
are  parallel  to  each  other  (35). 

The  intersections  of  three  planes  not  all  concurrent  are  par- 
allel to  each  other  (34). 

(2)  Lines  perpendicular : 

The  shortest  sect  from  a  point  to  a  plane  is  perpendicular  to 
all  lines  in  the  plane  through  its  foot  (51)  ;  or,  a  line  per- 
pendicular to  a  plane  is  perpendicular  to  all  lines  in  the 
plane  through  its  foot  (52). 

A  line  in  a  plane  and  perpendicular  either  to  a  line  or  to  its 
projection  upon  that  plane  at  their  point  of  intersection  is 
perpendicular  to  the  other  also  (84). 

(3)  Lines  concurrent : 

The  intersections  of  three  planes  meeting  in  pairs  are  con- 
current if  any  two  of  the  intersections  meet  (34). 

(4)  There  is  one,  and  but  one,  line 

through  a  given  point  perpendicular  to  a  given  plane  (53,  54) ; 
perpendicular  to  two  given  non-coplanar  lines  (87). 

(5)  To  construct  a  line 

through  a  given  point  perpendicular  to  a  given  plane  (86). 

(6)  The  projection 

upon  a  plane  of  a  line  oblique  or  parallel  to  that  plane  is  a 
straight  line  (78). 

(7)  Sects  equal  : 

Obliques  making  equal  angles  with  the  perpendicular  or  hav- 
ing equal  projections  upon  the  plane  are  equal  (79). 
Projections  of  equal  obliques  from  a  point  are  equal  (81). 

(8)  Sects  unequal : 

From   a  given  point   to   a  given   plane,   one    sect   is   the 

shortest  (50). 
The  perpendicular  from  a  point  to  a  plane  is  shorter  than 

any  oblique  from  that  point  (79). 


SUMMARY  OF   PROPOSITIONS  269 

Obliques  making  unequal  angles  with  the  perpendicular,  or 
having  unequal  projections  upon  the  plane  are  unequal  (79). 

Projections  of  unequal  obliques  from  a  point  to  a  plane  are 
unequal  (81). 

The  common  perpendicular  is  the  shortest  sect  between  two 
nou-coplanar  lines  (87). 

(9)  Sects  are  proportional : 

if  they  are  cut  by  three  parallel  planes  (48)  ; 

if  they  are  from  a  given  point,  cut  by  two  parallel  planes  (49). 

II.   THE  PLANE: 

(1)  Is  determined 

by  three  non-collinear  points  (5)  ; 
by  a  line  and  a  point  outside  (9)  ; 
by  two  intersecting  lines  (9)  ; 
by  two  parallel  lines  (9). 

(2)  There  is  one,  and  but  one,  plane 
through  a  given  point 

parallel  to  a  given  plane  (30,  36)  ; 
parallel  to  two  given  non-coplanar  lines  (46); 
perpendicular  to  a  given  line  (53)  ; 
through  a  given  line 

parallel  to  a  given  non-coplanar  line  (44)  ; 
perpendicular  to  a  given  plane  (unless  the  given  line  is 
itself  perpendicular  to  the  plane)  (76). 

(3)  Two  planes  are  parallel : 

when  they  have  no  point  in  common  (21)  ; 

if  one  contains  two  intersecting  lines  parallel  to  the  other 

m    (29); 

if  two  intersecting  lines  of  one  are  parallel  respectively  to 

two  intersecting  lines  of  the  other  (42)  ; 
if  they  are  parallel  to  the  same  plane  (37)  ; 
if  they  are  perpendicular  to  the  same  line  (56). 

(4)  Two  planes  are  perpendicular  : 

when  their  measuring  angle  is  a  right  angle  (70)  ; 
if  one  contains  a  line  perpendicular  to  the  other  (70). 


270  LINES   AND   PLANES 

(5)    To  construct  a  plane 
through  a  given  point 

parallel  to  a  given  plane  (32)  ; 

parallel  to  two  given  non-coplanar  lines  (47)  ; 

perpendicular  to  a  given  line  (63)  ; 

perpendicular  to  a  given  plane  (85)  ; 
through  a  given  line 

parallel  to  a  given  non-coplanar  line  (45). 

III.   LINES  AND  PLANES  :  t 

(1)  A  line  is  in  a  plane: 

if  two  of  its  points  are  in  the  plane  (6)  ; 

if  one  of  its  points  and  a  line  parallel  to  it  are  in  the  plane  (23) ; 

if  one  of  its  points  is  in  the  plane,  and  the  line  and  the 

plane  are  both 

parallel  to  the  same  line  (25)  ; 

parallel  to  the  same  plane  (26)  ; 

perpendicular  to  the  same  line  (60,  61)  ; 

perpendicular  to  the  same  plane  (73). 

(2)  Relative  positions 

of  two  lines:  (a)  intersecting,  (6)  parallel,  (c)  non-coplanar 

(13); 

of  a  line  and  a  plane:  (a)  the  line  in  the  plane,  (/>)  inter- 
secting, (c)  parallel  (14) ; 
of  two  planes:   (a)  coinciding,  (b)  intersecting,  (c)  parallel 

(15); 

of  three  planes  :  («)  intersecting  in  three  lines,  (b)  intersect- 
ing in  two  lines,  (c)  all  meeting  in  one  line,  (d)  mutually 
parallel  (33). 

(3)  Intersections 

of  a  line  and  a  plane,  —  a  point  (14)  ; 
of  two  planes,  —  a  line  (19)  ; 
of  three  planes : 

a  line  (38)  ; 

two  parallel  lines  (35)  ; 

three  lines  parallel  or  concurrent  (34). 


SUMMARY   OF   PROPOSITIONS  271 

(4)  A  line  is  parallel  to  a  plane  : 

if  they  have  no  point  in  common  (14)  ; 

if   it  is  not  in  the  plane  and  is  parallel  to  a  line  in  the 

plane  (24)  ; 
if  it  is  not  in  the  plane,  and  the  line  and  the  plane  are  both 

parallel  to  the  same  line  or  plane  (41). 

(5)  A  line  is  perpendicular  to  a  plane  : 

if  it  is  perpendicular  to  all  lines  in  the  plane  through  its 

foot  (52)  ; 
if   it  is   perpendicular   to    two    intersecting    lines    in    the 

plane  (62) ; 
if  it  is  perpendicular  to  a  plane   that   is  parallel  to  that 

plane  (57)  ; 

if  it  is  parallel  to  a  line  that  is  perpendipular  to  the  plane  (58)  ; 
if  it  is  in  a  plane  perpendicular  to  it  and  is  perpendicular 

to  the  intersection  of  the  two  planes  (72)  ; 
if  it  is  the  intersection  of  two  planes  that  are  perpendicular 

to  that  plane  (74). 

(6)    Transversals  of  parallels : 

If  a  line  cuts  one  of  two  parallel  planes,  it  cuts  the  other 

also  (28). 
If  a  plane  cuts  one  of  two  parallel  lines,  it  cuts  the  other 

also  (28). 
If  a  plane  cuts  one  of  two  parallel  planes,  it  cuts  the  other 

also  (36). 

IV.   ANGLES  : 

(1)  Equal  plane  angles : 

Angles  whose  arms  are  respectively  parallel  are  equal  (42). 
•    Angles  made  by  equal  obliques  with  a  perpendicular^are  equal 

(81). 
Measuring  angles  of  equal  dihedral  angles  are  equal  (66). 

(2)  Unequal  plane  angles: 

Angles  made  by  unequal  obliques  with  a  perpendicular  are 

unequal  (81). 
The  angle  between  an  oblique  line  and  its  projection  upon  a 

plane  is  the  least  angle  it  makes  with  any  line  in   the 

plane  (82). 


272  LINES  AND  PLANES 

(3)  Dihedral  angles 

having  equal  plane  angles  are  equal  (67)  ; 
are  proportional  to  their  plane  angles  (68)  ; 
are  measured  by  their  plane  angles  (69) . 

(4)  Measuring  angles  have  their  plane  determined , 

by  two  lines  in  the  faces  perpendicular  to  the  edge  at  the 

same  point  (65)  ; 
by  two  intersecting  lines  perpendicular  to  the  faces  of  the 

dihedral  angle  (75). 

V.   Locus: 

of  points  equidistant  from 

two  given  points,  —  a  plane  perpendicular  to  their  sect  at  its 

midpoint  (8ft)  ; 
three  given  non-collinear   points,  —  a   line   perpendicular   to 

their  plane  at  the  circuincenter  of  the  triangle  of  which 

they  are  vertices  (90)  ; 
four  given  non-coplanar  points  of  which  no  three  are  collinear, 

—  the  point  of  concurrence  of  the  planes  perpendicular  to 

their  sects  at  their  midpoint  (91)  ; 
two  intersecting  planes,  —  the  planes  bisecting  the  dihedral 

angles  formed  (92)  ; 
three  planes  meeting  in  a  point,  —  the  lines  of  intersection  of 

the  planes  bisecting  the  dihedral  angles  (93) ; 
four  planes  meeting  in  four  points,  —  the  points  common  to 

the  bisectors  of  the  dihedral  angles  (94). 

96.  ORAL   AND   REVIEW   QUESTIONS 

What  are  the  two  most  fundamental  statements  that  can  be  made 
about  a  plane  ?  In  order  to  use  plane  geometry  propositions  in  proving 
solid  geometry  theorems,  what  must  first  be  done  ?  By  what  methods 
can  this  be  done  ?  Show  that  two  transversals  of  the  arms  of  an  angle 
are,  in  general,  coplanar.  What  is  the  exception  ?  In  what  relative 
positions  can  a  point  and  a  line  lie?  a  point  and  a  plane ?  a  line  and 
a  plane  ?  two  lines  ?  two  planes  ?  What  is  the  intersection  of  a  line 
and  a  plane?  Why?  of  two  planes?  Why?  State  six  methods  of 
proving  that  a  line  is  in  a  plane.  Why  is  it  ever  necessary  to  prove 
that  a  line  is  in  a  plane  ?  What  is  the  definition  of  a  line  parallel  to 


ORAL   AND   REVIEW   QUESTIONS  273 

a  plane  ?  Is  there  a  simpler  method  of  showing  that  a  line  is  parallel 
to  a  plane?  What  is  the  definition  of  parallel  planes?  State  two 
other  methods  of  proving  planes  parallel,  founded  directly  on  this  defi- 
nition, and  state  and  prove  the  other  theorem  used  in  each  method. 
How  many  intersecting  lines  can  be  parallel  to  the  same  line?  parallel 
to  the  same  plane?  How  many  intersecting  planes  can  be  parallel  to 
.the  same  plane?  to  the  same  line?  State  all  possibilities  for  the 
relative  positions  of  three  planes.  If  three  planes  intersect  in  pairs,  is 
it  possible  for  them  to  have  two  non-coplanar  intersections  ?  Is  it  pos- 
sible with  four  planes?  Are  planes  necessarily  parallel  if  they  con- 
tain parallel  lines  ?  Are  lines  necessarily  parallel  if  they  are  in  parallel 
planes  ?  How  many  lines  through  a  point  can  be  perpendicular  to  the 
same  line  ?  to  the  same  plane  ?  How  many  intersecting  planes  can  be 
perpendicular  to  the  same  line?  to  the  same  plane? 

Which  of  the  following  must  be  parallel  (or  coincident,  including  as 
a  special  case  the  line  being  in  the  plane)  :  lines  parallel  to  the  same 
line ;  planes  parallel  to  the  same  plane ;  lines  parallel  to  the  same  plane  ; 
planes  parallel  to  the  same  line  ;  lines  perpendicular  to  the  same  line ; 
lines  perpendicular  to  the  same  plane ;  planes  perpendicular  to  the  same 
line ;  planes  perpendicular  to  the  same  plane  ?  How  many  intersect- 
ing lines  can  be  perpendicular  to  the  same  line?  to  the  same  plane? 
What  is  the  definition  of  perpendicular  planes?  How  many  lines  are 
used  to  show  the  planes  perpendicular  to  each  other  ?  Can  it  be  proved 
with  a  less  number?  If  two  planes  are  perpendicular  to  each  other, 
how  can  a  line  be  drawn  in  one  so  as  to  be  perpendicular  to  the 
other?  If  two  planes  are  perpendicular  to  each  other,  is  every 
line  in  one  perpendicular  to  the  other?  Are  all  perpendiculars  to 
one  plane  at  the  intersection  in  the  other  plane  ?  Where  is  the  pro- 
jection of  lines  in  one  plane  on  the  other  plane?  What  is  the  defi- 
nition of  a  line  perpendicular  to  a  plane?  How  many  lines  of  the 
plane  does  it  use  ?  Can  the  line  be  proved  perpendicular  to  the  plane 
with  fewer  lines  ?  State  two  methods  that  use  parallels  to  prove  a 
line  perpendicular  to  a  plane.  State  two  methods  that  use  perpen- 
dicular planes  to  prove  a  line  perpendicular  to  a  plane. 

Define  dihedral  angle,  and  its  plane  angle.  In  what  two  ways  is 
the  plane  of  a  measuring  angle  determined  ?  Explain  in  full  in  what 
sense  and  why  the  plane  angle  of  a  dihedral  angle  measures  it.  Can 
the  projection  of  a  straight  line  upon  a  plane  ever  be  a  curve?  Can 
the  projection  of  a  curve  ever  be  a  straight  line  ?  Where  must  a  pro- 


274  LINES   AND   PLANES 

jected  line  lie,  in  order  that  its  projection  upon  a  plane  shall  be  a  straight 
line  («)  if  it  is  straight?  (6)  if  it  is  not  straight?  If  a  line  when  pro- 
jected upon  each  of  two  planes  makes  straight  lines  on  both,  what  must 
be  true  of  it?  (Two  cases.)  Give  the  determining  lines  that  must  be 
drawn  in  order  to  construct  a  plane  (1)  through  a  point  (a)  parallel  to 
a  given  plane,  (ft)  parallel  to  two  non-coplanar  lines,  (c)  perpendicular 
to  a  line,  (d)  perpendicular  to  a  plane  ;  (2)  through  aline  (a)  parallel 
to  a  non-coplanar  line,  (ft)  perpendicular  to  a  plane.  State  two  methods 
that  have  been  used  to  prove  two  lines  perpendicular.  What  methods 
have  been  found  to  prove  angles  equal ?  unequal?  sects  equal?  un- 
equal ?  sects  proportional  ?  Upon  what  plane  geometry  locus  theorems 
can  the  following  be  made  to  depend  :  locus  of  points  equidistant  from 
two  points?  three  points?  two  lines?  two  planes?  State  all  the 
propositions  that  prove  that  there  can  be  a  line  or  a  plane  that  fulfills 
certain  conditions ;  all  propositions  that  prove  that  there  can  be  but 
one  line  or  one  plane  that  fulfills  certain  conditions. 

GENERAL  EXERCISES 

82.  If  two  perpendiculars  to  a  plane  from  two  points  on  the  same 
side  of  it  are  equal,  the  line  joining  those  points  is  parallel  to  the 
plane.     If  three  or  more  such  perpendiculars   are  equal,  the  points 
determine  a  plane  parallel  to  the  given  plane. 

83.  If   two  planes  are  drawn   perpendicular   respectively  to   two 
non-coplanar  lines,  their  intersection  is  perpendicular  to  any  plane 
parallel  to  the  given  lines. 

84.  Find  the  locus  of  points  in  a  given  plane,  equidistant  from  two 
given  planes 

NOTE.     Two  points  equidistant  from  a  plane,  on  the  same  perpen- 
dicular to  that  plane,  are  called  symmetric  with  regard  to  that  plane. 

85.  Prove  that  two  points  symmetric  to  a  plane  are  equidistant 
from  any  point  in  the  plane. 

86.  If  A  and  B  are  on  the  same  side  of  plane  M,  find  the  shortest 
path  between  A  and  B  that  includes  one  point  of  M.     (Use  the  point 
symmetric  to  either  A  or  B.}     This  is  simply  finding  the  shortest 
way  to  go  from  A  to  M  and  back  to  B.     It  is  the  path  that  a  ray  of 
light  travels  when  it  meets  a  mirror  and  is  reflected;  it  is  the  path 
followed  by  a  billiard  ball  striking  a  cushion  and  bounding  back,  or 


GENERAL   EXERCISES  275 

by  a  tennis  ball  bounding  from  the  ground,  —  unless,  in  either  of 
these  cases,  the  ball  is  affected  by  some  motion,  such  as  whirling, 
other  than  the  mere  rebound. 

87.  If  two  balls  are  in  positions  A  and  B  on  a  billiard  table,  show 
how  the  point  on  a  cushion  could  be  determined  so  that  if  A  strikes 
the  cushion  at  that  point,  it  will  rebound  to  B.     Show  how  a  point 
could  be  determined  so   that  A  would  strike  two  cushions  and  re- 
bound to  B',   three  cushions;    all  four  cushions.     Note  that  in  the 
last  cases  symmetry  is  being  used  in  regard  to  two  or  more  planes. 

88.  Find  the  shortest  path  in  two  intersecting  planes  between  a 
point  in  one  of  the  planes,  and  a  point  in  the  other  plane. 

89.  Find  the  shortest  path  in  the  surface  of  a  box  from  one  corner 
to  the  opposite  corner. 

90.  If   two  lines   in   one  of   two  intersecting  planes  make  equal 
angles  with  the  intersection,  they  make  equal  angles  with  the  other 
plane,  and  conversely. 

91.  Perpendiculars  from  a  point  to  a  set  of  parallel  lines  £re  co- 
planar. 

92.  The  arms  of  an  angle  are  equally  inclined  to  any  plane  through 
its  bisector. 

93.  The  arms  of  an  angle  are  equally  inclined  to  any  plane  through 
a  coplanar  line  perpendicular  to  its  bisector. 

94-   If  a  line  is  perpendicular  to  one  of  two  intersecting  planes,  the 
projection  upon  the  other  plane  is  perpendicular  to  the  intersection. 

95.  If  any  number  of  planes  perpendicular  to  the  same  plane  have 
a  common  point,  they  intersect  in  a  common  line. 

96.  If  a  plane  and  a  line  are  parallel,  a  plane  perpendicular  to  the 
line  is  also  perpendicular  to  the  plane.     Is  the  converse  also  true? 

97.  If  three  or  more  lines  are  drawn  from  points  on  one  of  two 
non-coplanar  lines  perpendicular  to  the  other  line,  the  sects  cut  off 
on  the  two  non-coplanar  lines   by  these  perpendiculars  are  propor- 
tional. 

98.  If  a  common  perpendicular  is  drawn  to  two  lines  that  are  either 
parallel  or  non-coplanar,  the  plane  perpendicular  to  that  common  per- 
pendicular at  its  midpoint  bisects  every  transversal  of  the  two  lines. 


276  LINES  AND  PLANES 

99.  If  two  congruent  rectangles  are  placed  so  that  they  have  an 
edge  in  common  but  are  not  coplanar,  their  common  edge  is  perpen- 
dicular to  two  of  the  planes  determined  by  other  edges,  and  is  paral- 
lel to  a  third  plane.     State  the  general  case. 

100.  When   will   lines  perpendicular   to  two   intersecting  planes 
meet?     Prove  ib. 

101.  Planes  perpendicular  to  intersecting  lines  meet.     What  can 
be  told  about  their  intersection  ? 

102.  Given  three  lines,  no  two  of  which  are  coplanar :  (a)  pass  a 
plane  through  one  of  them  so  as  not  to  be  parallel  to  either  of  the 
others ;  (6)  draw  a  line  through  all  three  of  them.     Can  a  line  be 
drawn  through  one  of  them  parallel  to  the  other  two?  a  plane? 

103.  Given  a  line  lying  within  and  parallel  to  the  edge  of  a  dihe- 
dral angle,  how  could  a  plane  be  passed  through  this  line,  intersect- 
ing the  faces  of  the  angle,  so  that  the  parallels  so  formed  would  be 
equidistant  from  the  given  line? 

104.  Given  a  line  outside  a  dihedral  angle  parallel  to  its  edge,  how 
could  a  plane  be  passed  through  this  line  meeting  the  faces  of  the 
dihedral  angle  so  that  the  distance  apart  of  the  parallel  intersections 
shall  equal  the  distance  of  one  of  them  from  the  given  line. 

105.  If  a  circle  is  inscribed  in  a  triangle,  the  lines  from  any  point 
in  the  perpendicular  erected  to  the  plane  of  the  circle  at  the  center 
to  the  points  of  contact  are  perpendicular  to  the  sides  of  the  triangle. 

106.  The  lines  from  any  point  in  the  perpendicular  to  a  plane  at 
the  incenter  of  a  given  triangle,  perpendicular  to  the  sides   of  the 
triangle,  meet  them  at  the  points  where  they  are  tangent  to  the  in- 
scribed circle. 

107.  If  in  each  of  two  intersecting  planes  two  lines  are   drawn 
parallel  to  the  intersection,  and  such  that  all  four  lines  are  the  same 
distance  from  the  intersection,  those  lines   determine  two  pairs  of 
parallel  planes. 

108.  If  four  lines  determine  more  than  four  planes,  the  lines  are 
either  all  concurrent  or  all  parallel. 

109.  If  two  pairs  of  parallel  planes  intersect,  the  other  two  planes 
determined  by  their  intersections  meet  in  a  line  parallel  to  the  given 
planes  and  to  their  intersections,  and  equidistant  from  each  opposite 
pair  of  intersections. 


GENERAL   EXERCISES  277 

110.  If  one  pair  of  parallel  planes  is  perpendicular  to  a  second 
pair  of  parallel  planes,  the  distance  between  one  pair  of  opposite  in- 
tersections equals  the  distance  between  the  other  pair  of  opposite 
intersections. 

111.  If  three  planes  meet  in  parallel  lines,  and  two  of  them  make 
equal  angles  with  the  third,  then  the  bisector  of  the  dihedral  angle 
between  those  two  planes  is  perpendicular  to  the  third  plane. 

112.  If  two  pairs  of  parallel  planes  intersect,  and  the  consecutive 
pairs  of  their  intersections  are  the  same  distance  apart,  the  diagonal 
planes  of  the  figure  formed  are  perpendicular  to  each  other. 

113.  If  two  parallel  planes  are  crossed  by  two  non-parallel  planes 
that  make  equal  angles  with  one  of  them, 

(a)  those  planes  make  equal  angles  with  the  other  of  the  parallel 
planes ; 

(6)  the  opposite  intersections  are  equidistant; 

(c)  the  non-parallel  planes  meet  in  a  line  parallel  to  the  given 
intersections,  and  equidistant  from  those  in  either  of  the  parallel 
planes ; 

(<^J  the  given  intersections  in  one  of  the  non-parallel  planes  are  the 
same  distance  apart  as  the  intersections  in  the  other  of  those  planes. 

(e)  State  and  prove  the  converse  of  (W). 

114-  If  two  triangles  (as  ABC  and  XYZ}  in  different  planes  are 
so  placed  that  each  corresponding  pair  of  sides  intersect  (as,  AB  and 
XY,  BC  and  YZ,  CA  and  ZF),  then  the  lines  through  their  corre- 
sponding vertices  (as  AX,  BY  and  CZ)  are  either  parallel  or  concur- 
rent. 


BOOK  VII.    POLYHEDRONS,  CYLINDERS, 

AND    CONES 

SECTION   I.     THE   PRISM  AND   THE   CYLINDER 

97.  Polyhedrons.  A  geometric  solid  bounded  by  planes 
is  'A  polyhedron.  The  polygons  that  bound  it  are  its  faces, 
their  intersections  are  its  edges,  and  the  intersections  of 
its  edges  are  its  vertices;  the  faces  taken  together  make 
up  its  surface.  The  area  of  its  surface  is  the  area  of  the 
polyhedron,  and  the  amount  of  space  that  it  occupies  is  the 
volume  of  the  polyhedron.  See  also  §  135,  which  explains 
how  volume  is  measured. 


A  POLYHEDRON  A  TETRAHEDRON 

98.  Number  of  Faces.  It. is  evident  that,  in  order  to 
inclose  space,  a  polyhedron  must  have  at  least  three  faces 
about  each  vertex,  and  at  least  four  faces  in  all.  A  solid 
bounded  by  four  faces  is  called  a  tetrahedron;  by  five 
faces,  a  pentahedron  ;  by  six  faces,  a  hexahedron  ;  by  eight 
faces,  an  octahedron;  by  ten  faces,  a  decahedron;  by 
twelve  faces,  a  dodecahedron;  and  by  twenty  faces,  an 
icosahedron.  The  corresponding  prefixes  are  used  for  the 
polyhedrons  of  any  number  of  faces,  but  those  here  men- 
tioned include  the  most  common. 

278 


THE  PRISM  AND  THE  CYLINDER 


279 


99.  Polyhedral  Angles.  When  three  or.  more  planes 
meet  in  a  point,  they  include  &  polyhedral  angle,  or  a  solid 
angle. 


In  the  figure,  the  planes  OVW,  OWX,  OXY,  OYZ,  OZV 
meet  in  O,  and  form  a  polyhedral  angle  O,  or  more  defi- 
nitely O-VWXYZ. 

The  common  point  O  is  its  vertex,  the  portions  of  planes 
that  bound  it  are  its  faces,  and  the  intersection  lines  of  its 
faces  are  its  edges.  The  faces  and  edges  extend  indefi- 
nitely from  the  vertex,,  but,  for  convenience  in  dealing 
with  a  polyhedral  angle,  a  plane,  as  ABODE,  is  sometimes 
considered  as  having  cut  its  edges. 

100.  Dihedral  and   Face  Angles.     A  polyhedral  angle 
evidently  has  as  many  dihedral  angles  as  edges,  and  as 
many  face  angles,  —  that  is,  angles  about  the  vertex  in  its 
faces,  as   Z  AOB,  /.BOG,  etc.,  —  as  faces.     The    dihedral 
angles  and  face  angles  are  sometimes  called  its  parts,  just  as 
the  sides  and  angles  of  a  triangle  are  referred  to  as  its  parts. 

A  polyhedron  has  a  polyhedral  angle  at  each  vertex, 
a  dihedral  angle  at  each  edge,  and  face  angles  about  each 
vertex,  the  number  of  face  angles  depending  on  the  kind 
of  polyhedral  angle  at  that  point. 

101.  Sections.     When  a  plane  is  passed  through  a  solid, 
the  figure  in  the  plane  bounded  by  its  intersection  with 


280         POLYHEDRONS,   CYLINDERS,   AND  CONES 

the  surface  of  the  solid  is  called  a  plane  section,  or  simply 
a  section,  of  the  solid.  It  is  evident  that  any  section  of  a 
solid  must  be  bounded  by  a  closed  line,  or  lines.  Why  ? 

Similarly,  a  plane  passed  through  all  the  edges  of  a 
pylyhedral  angle,  but  not  through  the  vertex,  forms  a 
polygon  of  as  many  sides  as  the  polyhedral  angle  has 
faces.  In  the  diagram  of  §  99,  ABODE  is  a  section  of  the 
polyhedral  angle  O. 

102.  Convex  Figures.     Either  a  polyhedral  angle,  or  a 
polyhedron,  is  said  to  be  convex  if  no  part  of  the  plane  of 
any  one  of  its  faces  lies  within  the  space  inclosed  by  its 
faces.     As  only  convex  figures  will  be  dealt  with  in  this 
book,    the    word    convex    will    be    understood   whenever 
polyhedral  angles  or  polyhedrons  are  used. 

103.  The  General  Polyhedron.     Elementary  geometry  is 
concerned  mostly  with  certain  classes  of  polyhedrons,  such 
as  prisms,  parallelepipeds,  and  pyramids,  all  of  which  will 
be  defined  when  they  are  to  be  used.     The  general  poly- 
hedron is  discussed  to  some  extent  in  §  323. 

104.  The  Prismatic  Surface  and  the  Cylindrical  Surface. 
If  a  moving  straight  line  always  remains  parallel  to  its 
original  position,  and  always  intersects  a  given  straight 
line,  it  evidently  describes,  or  generates,  a  plane.     Why  ? 

In  the  figures  on  p.  281,  AB  represents  the  original 
position  of  the  moving  line  (or  generatrix),  while  XY  rep- 
resents the  guiding  line  (or  directrix). 

A  straight  line  that  always  remains  parallel  to  its  origi- 
nal position,  and  always  intersects  a  line  that  is  in  one  plane, 
but  is  not  coplanar  with  the  moving  line, 

(a)  if  the  guiding  line  is  broken,  generates  a  prismatic 
surface;  which  is  evidently  composed  of  a  number  of  por- 


THE  PRISM   AND  THE   CYLINDER 


281 


PLANE 


PRISMATIC  SURFACE        CYLINDRICAL  SURFACE 


tions  of  planes  intersecting  in  parallel  lines,  which  are 
the  edges  of  the  prismatic  surface. 

(5)  if  the  guiding  line  is  curved,  generates  a  cylindrical 
surface,  of  which  no  part  is  plane,  for  if  it  were,  that  part 
of  the  guiding  line  would  be  straight.  The  moving  line 
in  each  of  its  various  positions  is  called  an  element  of  the 
cylindrical  surface.  For  some  purposes,  it  is  convenient 
to  let  "  element "  also  apply  to  prismatic  surfaces. 

A  non-coplanar  guiding  line  may  give  a  combination  of 
prismatic  and  cylindrical  surfaces,  and  need  not  be  con- 
sidered in  an  elementary  course. 

105.  Closed  Surfaces.  If  the  guiding  line  is  a  closed 
line,  the  prismatic  or  cylindrical  surface  is  also  closed,  and 
the  space  inclosed  is  called  prismatic  or  cylindrical  space. 


X 


U 


106.    Sections.     If  a  plane  that  is  not  parallel    to   the 
moving  line  meets  a  closed  prismatic  or  cylindrical  surface, 


282         POLYHEDRONS,   CYLINDERS,   AND   CONES 


it  cuts  all  elements  and  so  cuts  the  surface  in  a  closed  line. 
The  figure  bounded  by  this  closed  line  is  called  a  section 
of  the  prismatic  or  cylindrical  space.  If  it  is  perpendicu- 
lar to  an  edge  or  element,  it  is  a  right  section  ;  otherwise  it 
is  an  oblique  section.  When  two  sections  are  made  by 
parallel  planes,  they  are  called  parallel  sections.  In  the 
figure,  ABCD  and  EFGH  are  parallel  sections  of  the  pris- 
matic space  ;  XT  is  a  section  of  the  cylindrical  space. 

*  107.    The  edges   of  a  prismatic  surface,    or   the  ele- 
ments of  a  cylindrical  surface,  between  parallel  sections 
are  equal. 

*  108.    Parallel  sections  of  a  prismatic  space  are  con- 
gruent polygons. 

109.    Theorem  I.      Parallel  sections  of  a  cylindrical 
space  are  congruent. 


Let  planes  M  and  N  make  parallel  sections  of  a  cylin- 
drical space.  Take  points  A  and  B  on  the  perimeter  of 
section  3f,  and  let  P  represent  any  third  point  of  that 
perimeter.  Draw  the  elements  through  A,  B,  and  P  to  Af, 
#',  and  P'  on  the  perimeter  of  section  jy,  and  draw 


THE   PRISM  AND   THE   CYLINDER  283 

AB,  AP,  BP,  A'B',  A'P',  B'P'I  Then  AABP^AAfBrPf 
(why  ?),  and  if  section  M  is  superposed  on  section  N  with 
AB  coinciding  with  A'B',  P  falls  on  p1 .  But  P  may  be 
any  point  on  the  perimeter  of  section  M,  so  each  point  of 
section  M  falls  on  a  corresponding  point  of  section  N,  and 
conversely,  and  the  two  sections  coincide  and  are  congruent. 

110.  Prisms  and  Cylinders.     The  part  of  a  prismatic 
space   between  two  parallel  sections   is  called   a  prism. 
The  part   of   a   cylindrical   space   between  two   parallel 
sections  is  called  a  cylinder.     The  parallel  sections  are 
the  bases,  while  the  prismatic  or  cylindrical  surface  is  the 
lateral  surface  of  the  figure.     The  area  of  the  lateral  sur- 
face is  the  lateral  area  of  the  figure.     In  the  case  of  the 
prism,  the  planes  forming  the  lateral  surface  are  lateral 
faces ;  their  intersections  are  lateral  edges.     A  prism  or  a 
cylinder  is  right  or  oblique  according  as  its  bases  are  right 
or  oblique  sections. 

111.  Altitude.     The  altitude  of  a  prism  or  a  cylinder  is 
the  perpendicular  between  its  bases. 

112.  Classification   of   Prisms   according   to  Number   of 
Faces.    A  prism  is  triangular,  quadrangular,  pentagonal,  etc., 
according  as  its  bases  have  three,  four,  five,  etc.,  sides. 

113.  Regular  Prisms.     A  right  prism  whose  bases  are 
regular  polygons  is  a  regular  prism. 

114.  Circular  Cylinders.     A  cylinder  whose  bases  are 
circles  is  a  circular  cylinder;    if  its  bases  are  also  right 
sections,  it  is  a  right  circular  cylinder,  or  since  it  might 
be  generated  by  revolving  a  rectangle  around  one  side  as 
an  axis,  it  is  sometimes   called  a  cylinder  of  revolution. 
The  cylinders  considered  in  elementary  solid  geometry  are 
almost  always  circular  cylinders.     See  Appendix,  §  321. 


284         POLYHEDRONS,   CYLINDERS,   AND   CONES 

115.  Axis  and  Radius  of»a  Circular  Cylinder.  The  line 
joining  the  centers  of  the  bases  of  a  circular  cylinder  is 
called  its  axis.  The  radius  of  the  base  of  a  circular 
cylinder  is  called  the  radius  of  the  cylinder. 

115.  Find  the  locus  of  points  at  a  fixed  distance  from  a  given  line. 

116.  The  Cylinder  the  Limiting  Case  of  a  Prism.     A 

prism  is  said  to  be  inscribed  in,  or  circumscribed  about  a 
cylinder  if  its  bases  are  respectively  inscribed  in,  or  cir- 
cumscribed about,  the  bases  of  the  cylinder.  The  follow- 
ing statement  is  assumed  without  proof.  Its  truth  is 
manifest,  but  it  is  difficult  to  prove. 

Theorem  II.  If  a  prism,  the  base  of  which  is  a  regu- 
lar polygon,  is  inscribed  in,  or  circumscribed  about,  a 
given  circular  cylinder,  the  lateral  area  of  the  prism 
approaches  the  lateral  area  of  the  cylinder  as  its 
limit,  and  the  volume  of  the  prism  approaches  the 
volume  of  the  cylinder  as  its  limit,  as  the  number  of 
faces  of  the  prism  is  increased  indefinitely. 

It  follows  that  the  circular  cylinder  is  the  limiting 
case  of  the  prism  having  a  regular  base,  and  that 
theorems  true  for  the  one  will  usually  be  true  for  the 
other.  This  is  why  the  two  solids  are  here  treated  to- 
gether. 

117.  Theorem  III.     A  section  of  a  cylinder  made  by  a 
plane  containing  an  element  is  a  parallelogram. 

116.  A  section  of  a  prism  made  by  a  plane  parallel  to  an  edge  is  a 
parallelogram. 

118.  Tangent  to  a  Cylinder.     A  plane  is  tangent  to  a 
cylinder  if  it  meets  its  surface  in  one  element,  and  in  no 
point  outside  that  element. 


THE   PRISM  AND   THE  CYLINDER  285 

*  119.    If  a  plane  is  tangent  to  a  circular  cylinder,  its 
intersection  with  the  planes  of  its  bases  are  tangent  to 
the  bases. 

For  the  intersection  has  but  one  point  on  the  circum- 
ference. 

*  120.     The  plane  determined  by  a  tangent  to  a  base  of 
a  circular  cylinder,  and  the  element  from  the  point  of 
contact,  is  tangent  to  the  cylinder. 

For  if  it  met  the  cylinder  in  a  point  outside  the  ele- 
ment what  would  follow  ? 

121.  Theorem  IV.     The  lateral  area  of  a  prism  is  the 
product  of  an  edge  and  the  perimeter  of  a  right  section. 

What  form  would  this  take  for  a  right  prism  ? 

122.  COR.     The    lateral    area   of    a    right    circular 
cylinder  is  the  product  of  an  element  and  the  circum- 
ference of  the  base. 

Inscribe  a  regular  prism  of  lateral  area  _L',  edge  e,  and 
perimeter  of  the  base  p,  in  the  right  circular  cylinder  of 
lateral  area  L,  element  e,  and  circumference  of  the  base  c. 
Then  L'  =  ep.  If  the  number  of  faces  of  the  prism  is 
increased  without  limit,  L  =  Lf,  and  p  =  c.  Why  ?  But 
if  a  variable  approaches  a  limit,  its  product  by  a  constant 
will  approach  the  product  of  its  limit  by  that  constant, 
therefore  ep  =  ec.  Also,  since  L1  and  ep  are  two  variables 
that  are  constantly  equal,  and  are  approaching  the  limits  L 
and  ec,  those  limits  are  equal,  that  is,  L  =  ec.  If  the  cylin- 
der is  of  radius  r  and  altitude  A,  its  lateral  area  =  2  Trrh. 

117.  Prove,  without  using  the  formula  for  the  lateral  area,  that 
the  lateral  areas  of  two  prisms  cut  from  the  same  prismatic  space 
and  having  an  equal  lateral  edge  are  equivalent. 


286          POLYHEDRONS,   CYLINDERS,   AND   CONES 

118.  Prove,  without  using  the  formula  for  the  lateral  area,  that 
the  lateral  areas  of  two  cylinders  cut  from  the  same  cylindrical  space 
and  having  an  equal  element  are  equivalent. 

119.  Find  the  lateral  area  of  a  right  prism  of  altitude  5  in.,  and 
having  an  equilateral  triangle  of  side  4  in.  as  its  base. 

120.  Find  the  total  area  (lateral  area  and  bases)  of  a  regular  hex- 
agonal prism  of  edge  (base  and  lateral)  6  in. 

121.  Find  the  lateral  area  of  an  oblique  prism  of  edge  10  ft.,  hav- 
ing as  its  right  section  a  square  of  side  2.5  ft. 

122.  What  is  the  lateral  area  of  a  right  circular  cylinder  of  ele- 
ment 12  in.  and  base  of  radius  3  in.? 

123.  The  pillars  of  a  colonial  house  are  regular  hexagonal  prisms 
of  base  edge  1  ft.,  and  height  30  ft.     How  much  surface  must  be 
painted  on  four  of  them  ? 

124-  How  much  is  the  surface  of  a  log  14  in.  in  diameter  and  12£ 
ft.  long  diminished  by  cutting  the  largest  possible  square  timber  out 
of  it? 

125.  The  circular  right  section  of  a  cylinder  has  an  area  of  54 
sq.  ft.,  its  altitude  is  25  ft.,  and  its  elements  make  a  60°  angle  with 
the  base.     Find  the  lateral  area. 

123.  Theorem  V.     Two  right  prisms,  or  two  right  cylin- 
ders, are  congruent  if  they  have  congruent  bases  and 
equal  altitudes.     Superpose. 

124.  Equivalence.     Two  solids  are  said  to  be  equivalent 
when  their  surfaces  inclose  the  same  amount  of  space. 

126.  Two  cylinders  cut  from  the  same  space  and  having  an  equal 
element  are  equivalent.     Show  that  the  solid  from  the  lower  base  of 
one  to  the  lower  base  of  the  other  can  slide  along  the  cylindrical  space 
so  as  to  coincide  with  the  solid  between  the  upper  bases,  etc. 

127.  An  oblique  cylinder  is  equivalent  to  a  right  cylinder  whose 
base  is  its  right  section,  and  whose  altitude  is  its  element. 

125.  The   Parallelepiped.       A    prism   whose   bases   are 
parallelograms  is  called  a  parallelepiped.     It  is  manifest 
from  this  definition  that  all  the  faces  of  a  parallelepiped 


THE   PRISM   AND   THE   CYLINDER  287 

are  parallelograms,  and  that  it  is  therefore  bounded  by 
three  pairs  of  parallel  planes,  and  has  three  sets  of  four 
equal  parallel  edges.  Explain  in  full. 

*  126 .  Any  pair  of  opposite  faces  of  a  parallelepiped 
may  be  considered  to  be  its  bases.  For  the  parallelepiped 
fulfills  the  conditions  of  the  definition  of  a  prism  whichever 
faces  are  used  as  bases.  Therefore,  the  opposite  faces  of  a 
parallelepiped  are  congruent. 

127.  Right  and  Rectangular  Parallelepipeds.     A  parallel- 
epiped whose  lateral  edges  are  perpendicular  to  its  bases  is, 
by  the  definition  of  a  right  prism,  a  right  parallelepiped  ; 
otherwise  it  is  oblique.     If,  in  addition  to  being  right,  it 
has  rectangular  bases,  it  is  called  a  rectangular  parallel- 
epiped, for   all    its   faces   are    rectangles.     Many   shorter 
names  for  rectangular  parallelepipeds  have  been  suggested 
but  no  one  is  in  general  use.     The  term  cuboid  seems  as 
convenient  as  any. 

128.  Cube.     A    rectangular    parallelepiped   having  all 
its  edges  equal,  and  therefore  all  its  faces  square,  is  called 
a  cube. 

129.  Diagonal.     The  sect  joining  any  vertex  of  a  par- 
allelepiped to  the  one  not  in  the  same  face  is  called  a 
diagonal  of  the  parallelepiped. 

130.  Dimensions.      In  a  rectangular  parallelepiped  the 
lengths  of  the  three  edges  drawn  from  one  vertex  are  the 
dimensions  of  the  figure. 

131.  Ratio  between  Solids.     By  "the  ratio  of  one  solid 
to  another  "  is  meant  "  the  ratio  of  the  amounts  of  space 


288          POLYHEDRONS,   CYLINDERS,   AND   CONES 

occupied  by  them."     This  can  of  course,  be  worked  out 
only  in  terms  of  congruent  parts  of  the  solids. 

NOTE.  At  the  discretion  of  the  teacher,' §§  132,  133,  and  134,  may  be 
omitted  and  §  136  may  be  assumed  as  the  definition  of  the  volume  of  a  rec- 
tangular parallelepiped.  The  author  does  not  recommend  this  course,  but 
it  has  the  sanction  of  some  authorities. 

132.  Theorem   VI.       Two  rectangular  parallelepipeds 
having  two  dimensions  in  common  are  to  each  other 
as  their  third  dimensions, 

For  the  commensurable  case,  divide  the  third  dimen- 
sions by  a  common  divisor,  pass  planes  through  the  points 
of  division  dividing  the  given  parallelepipeds  into  con- 
gruent parts,  and  prove  the  proportion. 

The  incommensurable  case  can  be  proved  by  limits,  but 
it  is  not  of  Gen  required  in  an  elementary  course. 

133.  Theorem  VII.     Two  rectangular  parallelepipeds 
having  one  dimension  in  common  are  to  each  other  as 
the  products  of  their  other  dimensions. 

Compare  each  with  a  third  parallelepiped  so  con- 
structed as  to  make  this  possible. 

134.  Theorem  VIII.     Two  rectangular  parallelepipeds 
are  to  each  other  as  the  products  of  their  three  dimen- 
sions.   Use  the  method  of  Theorem  VII. 

128.  Prove  Theorem  VIII  without  using  Theorem  VII,  by  con- 
structing two  auxiliary  parallelepipeds  with  which  to  compare  the 
given  ones. 

135.  Volume:  Cubic  Unit.     The  unit  of  measure  for  solid 
figures  is  the  cubic  unit,  or  unit  of  volume,  which  is  a  cube 
whose  edge  is  a  linear  unit,  and  whose  faces  are  therefore 
units  of  area.     Since  volume  is  always  expressed  in  cubic 
units,  the  volume  of  any  solid  is  its  ratio  to  a  cubic  unit, 
or,  in  other  words,  the  number  of  cubic  units  it  contains. 


THE   PKISM   AND   THE   CYLINDER  289 

136.  Theorem  IX.    The  volume  of  a  rectangular  par- 
allelepiped is  the  product  of  its  three  dimensions. 

Let  B  be  a  parallelepiped  of  dimensions  a,  5,  c,  and  let 
U  be  a  unit  of  volume,  of  edges  equal  to  the  linear  unit  u. 
Then  compare  the  two  parallelepipeds  : 

E      a      b       c 
—  =  -  X  -  X  -, 
U     u      u      u 

But  —  is  the  volume  of  the  parallelepiped,  and  -,  -,  and  - 
U  u   u  u 

are  the  three  dimensions  in  length  units,  therefore, 

The  volume  of  a  rectangular  parallelepiped  (in  cubic 
units)  equals  the  product  of  its  three  dimensions  (in  length 
units).  Evidently  this  volume  might  also  be  expressed 
as  the  product  of  its  base  and  its  altitude.  Why  ? 

129.  How  many  cubic  yards  of  earth  must  be  removed  in  digging  a 
rectangular  ditch  6  ft.  deep,  3  ft.  wide,  and  one  quarter  of  a  mile  long  ? 

130.  A  log  2  ft.  in  diameter  and  24  ft.  long  has  been  sawed  in  two 
lengthwise,  making  two  semicircular  timbers.     If  the  largest  possible 
square  timber  is  cut  out  of  each  of  these  parts,  how  many  board  feet 
(1  ft.  square  and  1  in.  thick)  will  they  contain  ? 

131.  In  ex.  130,  if  the  largest  possible  square  log  had  been  cut 
from  the  original  log,  how  would  the  number  of  board  feet  in  it 
compare  with  the  number  in  the  two  timbers  ? 

137.  Theorem  X.     If  two  solids  contained  between  the 
same  two  parallel  planes  are  such  that  their  sections  by 
any  third  plane  parallel  to  these  two  planes  are  equiva- 
lent, the  two  solids  have  the  same  volume. 

This  is  "  Cavalieri's  Theorem,"  and  such  solids  as  those 
described  are  called  Oavalieri  bodies.  The  theorem  de- 
pends on  limits  for  its  proof,  but  the  proof  is  too  difficult 
at  this  point.  See  Appendix,  §  319. 

To  prove  two  solids  equivalent  in  volume  by  this  theo- 


290         POLYHEDRONS,   CYLINDERS,   AND   CONES 

rem,  it  is  necessary  (1)  to  show  that  the  solids  can  be 
placed  between  the  same  parallel  planes ;  (2)  to  pass  any 
third  parallel  plane  through  the  solids,  and  to  prove  that 
the  sections  obtained  are  equivalent. 

138.  Theorem   XI.     Prisms   cut  from  the  same  pris- 
matic space  and  having  equal  lateral  edges  are  equiva- 
lent in  volume. 

For  the  part  of  the  prismatic  space  between  their  upper 
.bases  can  coincide  with  the  part  between  their  lower  bases  ; 
and  after  taking  each  of  these  parts  from  the  whole  figure 
formed  by  the  prisms,  equivalent  prisms  remain.  Draw 
the  prisms  so  that  their  bases  do  not  intersect,  as  this 
would  unnecessarily  complicate  the  figure.  They  may  be 
entirely  outside  each  other^  or  may  have  some  common 
space.  Note  the  similarity  between  this  proof  and  one 
method  of  proving  that  parallelograms  having  equal  bases 
and  lying  between  parallels  are  equivalent. 

139.  Theorem    XII.     A   parallelepiped   is  equivalent 
to  a  rectangular  parallelepiped  of  equivalent  base  and 
equal  altitude. 


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For,  given  the  oblique  parallelepiped  B,  with  the  base 
ABCD,  and  the  altitude  h,  pass  two  planes  M  and  N,  whose 
distance  apart,  XF,  equals  AB,  through  the  prismatic  space 
of  which  E  is  a  part,  so  as  to  be  perpendicular  to  AB. 


THE   PRISM   AND   THE   CYLINDER  291 

Then  these  planes  cut  from  the  prismatic  space  a  new 
parallelepiped  S  equivalent  to  .R,  since  XY  =  AB,  and 
having  its  rectangular  base  XTZW  equivalent  to  ABCD 
(why?),  and  the  same  altitude  h.  In  S,  the  planes  M 
and  N  are  perpendicular  to  XYZW,  but  the  other  lateral 
faces  may  or  may  not  be  perpendicular  to  the  base. 

The  same  operation  applied  to  S  by  extending  FZ  and 
its  parallels  to  form  the  edges  of  a  prismatic  space,  and 
passing  planes  perpendicular  to  YZ  and  the  distance  FZ 
apart,  would  form  a  third  parallelepiped  equivalent  to 
8  and  therefore  to  #,  and  having  a  base  equivalent  to 
the  bases  of  those  figures,  and  the  same  altitude  h. 
But  this  parallelepiped  would,  by  construction,  be  rectan- 
gular, for  its  base  would  be  a  rectangle,  and  the  lateral 
planes  would  all  have  been  constructed  perpendicular  to 
the  base.  Therefore  it  is  possible  to  transform  any  paral- 
lelepiped into  an  equivalent  rectangular  parallelepiped 
with  an  equivalent  base  and  the  same  altitude. 

But  one  of  the  transformations  is  shown  in  this  figure, 
because  the  attempt  to  repeat  the  operation  in  the  same 
figure  makes  an  unnecessarily  complicated  diagram. 

Alternative  method  :  Use  Cavalieri's  Theorem. 

140.  COR.     The  volume  of  any  parallelepiped  is  the 
product  of  its  base  and  its  altitude. 

141.  Theorem    XIII.      A  plane  through    two  opposite 
edges  of  a  parallelepiped  divides  it  into  equivalent  tri- 
angular prisms. 

If  the  parallelepiped  is  right,  the  prisms  are  congruent. 
If  not,  it  is  equivalent  to  a  right  parallelepiped  cut  from 
the  same  space,  and  the  parts  into  which  the  plane  cuts  it 
are  equivalent  to  the  halves  of  that  right  parallelepiped. 


292         POLYHEDRONS,   CYLINDERS,   AND  CONES 

142.  COR.    1.     The  volume  of  a  triangular  prism  is 
the  product  of  its  base  and  its  altitude* 

143.  COR.    2.     The  volume  of  any  prism  is  the  prod- 
uct of  its  base  and  its  altitude. 

144.  COR.    3.     The  volume  of  a  circular  cylinder  is 
the  product  of  its  base  and  its  altitude.     For  it  is  the 
limit  approached  by  the  inscribed  prism  of  regular  base 
when  the  number  of  faces  is  increased  indefinitely.     (See 
§  116.)     The  volume  of  a  cylinder  of  radius  r  and  altitude 
h  is  7rr2h. 

182.  Which  would  be  lighter  and  how  much,  circular  pillars  23  in. 
in  diameter  and  12  ft.  high,  or  octagonal  pillars  24  in.  in  diameter  and 
12  ft.  high,  if  they  were  made  of  cement  weighing  150  pounds  to  the 
cubic  foot? 

133.  What  per  cent  of  the  lumber  is  cut  off  in  cutting  a  square 
timber  out  of  a  log  of  radius  r?     in  cutting  out  a  hexagonal  timber  ? 

134.  The  diagonals  of  a  parallelepiped  are  concurrent  in  the  mid- 
point of  each. 

135.  Any  section  of  a  parallelepiped  through  two  pairs  of  parallel 
planes  is  a  parallelogram. 

136.  If  the  diagonals  of  the  faces  of  a  parallelepiped  are  drawn,  the 
lines  joining  the  intersection  points  of  the  diagonals  in  opposite  faces 
are  concurrent  in  the  midpoint  of  each. 

137.  The  diagonal  of  a  cube  equals  the  square  root  of  three  times 
its  edge. 

138.  The  sum  of  the  squares  on  the  diagonals  of  a  parallelepiped 
equals  the  sum  of  the  squares  on  its  edges. 

139.  The  square  on  a  diagonal  of  a  rectangular  parallelepiped 
equals  the  sum  of  the  squares  on  its  three  edges. 

140.  Find  the  volume  of  a  circular  cylinder  of  radius  5  cm.,  whose 
axis  is  1.5  m.  long,  and  makes  an  angle  of  45°  with  its  bases. 

141.  An  irrigation  ditch  is  4  ft.  wide,  3  ft.  deep  for  its  whole  width, 
with  a  semicylindrical  bottom  below  this  3  ft.    If  it  is  filled  to  within 
6  in.  of  the  top,  how  much  water  does  it  contain  to  the  mile  ? 


SECTION   II.     THE   PYRAMID   AND   THE   CONE 

145.  The  Pyramidal  Surface  and  the  Conical  Surface. 
If  a  moving  straight  line  always  contains  a  fixed  point, 
and  always  intersects  a  given  straight  line,  it  generates 
a  portion  of  a  plane.  Why  ? 


PLANE  SURFACE 


PYRAMIDAL,  SURFACE 


CONICAL  SURFACE 


In  these  figures,  AB  represents  the  original  position  of 
the  moving  line  (or  generatrix),  P  being  the  fixed  point. 
XY  represents  the  guiding  line  (or  directrix). 

A  moving  straight  line  that  always  contains  a  fixed 
point,  and  always  intersects  a  line  that  lies  entirely  in  one 
plane,  but  is  not  coplanar  with  the  moving  line : 

(a)  if  the  guiding  line  is  broken,  generates  a  pyramidal 
surface,  which  is  evidently  composed  of  portions  of  a  num- 
ber of  planes  having  one  common  point,  with  each  succes- 
sive pair  intersecting  in  lines  that  are  the  edges  of  the 
surface. 

(£>)  if  the  guiding  line  is  curved,  generates  a  conical  sur- 
face, of  which  no  part  is  plane,  for  if  it  were,  that  part  of 
the  guiding  line  would  be  straight.  (Why?)  The  mov- 

293 


294         POLYHEDRONS,   CYLINDERS,   AND   CONES 


ing  line  in  each  of  its  various  positions  is  called  an  element 
of  the  conical  surface. 

Non-coplanar  guiding  lines  may  give  combinations  of 
pyramidal  and  conical  surfaces,  and  need  not  be  con- 
sidered in  an  elementary  course. 

146.  Vertex.     The  fixed  point  through  which  the  mov- 
ing line  passes  is  called  the 

vertex. 

147.  Closed  Surfaces.    If 

the  guiding  line  is  a  closed 
line,  the  pyramidal,  or  coni- 
cal, surface  also  is  closed, 
and  the  space  inclosed  is 
called  pyramidal,  or  conical,  space  respectively. 

148.  Sections.     If  a  plane  cuts  all  the  edges  of  a  closed 
pyramidal  surface,  or  all  the  elements  of  a  closed  conical 
surface,  but  not  at  the  vertex,  it  cuts  the  surface  in  a 
closed  line,  and  the  figure  bounded  by  this  closed  line  is 
called  a  section  of  the  pyramidal  or  the  conical  space. 

When  two  sections  are  made  by  parallel  planes,  they 
are  called  parallel  sections. 

Sj  and  sz  are  parallel  sections  of  the  closed  pyramidal 
space  P,  and  of  the  closed  conical  space  O.  In  each  case 
F  is  the  vertex.  In  the  pyramidal  space  the  section  must 
be  a  polygon.  Why  ? 

*149.  The  edges  from  the  vertex  of  a  pyramidal  sur- 
face, or  the  elements  from  the  vertex  of  a  conical  surface, 
are  cut  proportionally  by  parallel  sections. 

150.  Theorem  XIV.  Parallel  sections  of  a  pyramidal 
space  are  similar  polygons,  and  their  areas  are  propor- 
tional to  the  squares  of  their  distances  from  the  vertex. 


THE  PYRAMID  AND  THE  CONE        295 

142.  Which  are  the  corresponding  points  on  the  perimeters  of 
parallel  sections?  If  a  line  is  drawn  from  the  vertex  of  a  conical 
space  through  two  parallel  sections,  show  that  the  sects  from  the  points 
thus  obtained  to  any  two  corresponding  points  on  the  perimeters  of 
the  sections  will  be  proportional  to  the  distances  of  those  sections 
from  the  vertex. 

151.  Pyramids  and  Cones.     That  part  of  a  pyramidal 
space  between  the  vertex  and  a  section  is  called  a  pyramid. 
That  part  of  a  conical  space  between  the  vertex  and  a 
section  is  called  a  cone.     The  section  is  called  the  base, 
and  the  pyramidal  or  conical  surface  is  called  the  lateral 
surface  of  the  figure,  while  the  area  of  the  lateral  surface 
is  called  the  lateral  area.     In  the  case  of  the  pyramid,  the 
portions  of  planes  forming  the  lateral  surface  are  called 
lateral  faces,  and  their  intersections  are  called  lateral  edges. 

152.  Truncated  Figures.     That  part  of  a  prismatic  or 
cylindrical  space  between  two  non-parallel  sections  that 
do  not  intersect  on  the  surface  is  called  a  truncated  prism 
or  cylinder. 

That  part  of  a  pyramidal  or  conical  space  between  two 
sections  on  the  same  side  of  the  vertex  that  do  not  inter- 
sect on  the  surface  is  called  a  truncated  pyramid  or  cone. 

153.  Frustums.     A  truncated  pyramid  or  cone  between 
parallel  sections  is  called  a  frustum  of  the  pyramid  or  cone. 

154.  Lateral  Faces  and  Edges.     The  definition  of  lateral 
surface,  lateral  faces,  and  lateral  edges  given  in  §  151  holds 
also  for  frustums  and  truncated  figures.     The  parallel  sec- 
tions in  a  frustum  are  its  bases,  but  only  one  of  the  sections 
is  in  general  considered  the  base  of  the  truncated  figure. 

155.  Altitude.     The  altitude  of  a  pyramid  or  a  cone  is 
the  perpendicular  from  the  vertex  to  the  base.      In  a  frus- 


296         POLYHEDRONS,   CYLINDERS,   AND  CONES 

turn,  it  is  the  perpendicular  between  the  bases.  A  trun- 
cated figure  cannot  be  said  to  have  an  altitude  ;  a  triangular 
truncated  prism,  however,  is  sometimes  said  to  have  three 
altitudes,  namely,  the  perpendiculars  from  the  three  ver- 
tices of  one  section  to  the  other  section. 

156.  Classification  of  Pyramids  according  to  Number  of 
Faces.     A    pyramid,    like   a   prism,    is   triangular,    quad- 
rangular, pentagonal,   etc.,    according   to   the  number  of 
sides  of  its  base. 

157.  Regular  Pyramids.     A  pyramid  whose  base  is  a 
regular  polygon,  and  whose  altitude  meets  the  base  at 
its  center,  is  called  a  regular  pyramid.     The  altitude  of 
such  a  pyramid  is  called  its  axis.     The  lateral  faces  of 
a  regular  pyramid  are  congruent.    (Why?).     The  altitude 
of  one  of  the  lateral  faces  is  called  the  slant  height  of 
the  pyramid. 

158.  Circular  Cones.     A  cone  whose  base  is  a  circle  is 
a  circular  cone.     The  line  joining  the  vertex  of  a  circular 
cone  to  the  center  of  its  base  is  called  the  axis  of  the  cone, 
and  if  the  axis  is  perpendicular  to  the  base,  the  cone  is  a 
right  circular  cone.     The  radius  of  the  base  is  called  the 
radius  of  the  cone.     The  cones  considered  in  elementary 
solid  geometry  are  usually  circular  cones.      See  Appendix, 
§321. 

The  right  circular  cone,  since  it  might  be  generated  by 
revolving  a  right  triangle  about  one  leg  as  an  axis,  is 
sometimes  called  a  cone  of  revolution.  An  element  of  a 
right  circular  cone  is  sometimes  called  its  slant  height. 

143.  Find  the  locus  of  a  point  such  that  its  distances  from  a  given 
plane  and  a  line  perpendicular  to  that  plane  shall  have  a  given  fixed 
ratio. 


THE  PYRAMID  AND  THE  CONE       297 

159.  The  Cone  the  Limiting  Case  of  the  Pyramid.    A  pyr- 
amid is  said  to  be  inscribed  in  or  circumscribed  about  a 
cone  if  the  pyramid  and  cone  have  the  same  vertex  and 
the  base  of  the  pyramid  is  respectively  inscribed  in  or  cir- 
cumscribed about  the  base  of  the  cone.      The  following 
theorem  is  assumed  without  proof : 

Theorem  XV.  If  a  pyramid,  the  base  of  which  is  a 
regular  polygon,  is  inscribed  in  or  circumscribed  about 
a  given  circular  cone,  the  lateral  area  of  the  pyramid 
approaches  the  lateral  area  of  the  cone  as  its  limit,  and 
the  volume  of  the  pyramid  approaches  the  volume  of  the 
cone  as  its  limit,  as  the  number  of  sides  of  the  base  of 
the  pyramid  is  increased  indefinitely. 

The  cone  is  then  a  limiting  case  of  the  pyramid ;  there- 
fore the  two  figures  will  be  treated  together. 

160.  Theorem  XVI.     A   section  of  a  cone  made  by  a 
plane  containing  an  element  is  a  triangle. 

144.  In  Theorem  XVI,  would  it  be  sufficient  to  say  "by  a  plane 
through  the  vertex  "  ? 

145.  A  section  of  a  pyramid  made  by  a  plane  through  the  vertex 
is  a  triangle. 

161.  Theorem  XVII.     A.  section  of  a  circular  cone  made 
by  a  plane  parallel  to  the  base  is  a  circle,  the  center  of 
which  is  the  point  where  the  axis  meets  it. 

Show  that  the  sects  from  this  point  to  the  perimeter  of 
the  sections  are  in  proportion  to  the  radii  of  the  base. 

162.  Tangant  to  a  Cone.     A  plane  is  tangent  to  a  cone  if 
it  meets  its  surface  in  one  element,  and  in  no  other  point. 

*  163.    If  a  plane  is  tangent  to  a  circular  cone,  its  inter- 
section with  the  plane  of  the  base  is  tangent  to  the  base. 


298         POLYHEDRONS,   CYLINDERS,   AND  CONES 

*  164.  The  plane  determined  ~by  a  tangent  to  the  base 
of  a  circular  cone  and  the  element  from  the  point  of 
contact,  is  tangent  to  the  cone. 

165.  Theorem  XVIII.     The  lateral  area  of  a  regular 
pyramid  is  equal  to  one  half  the  product  of  the  slant 
height  and  the  perimeter  of  the  base. 

166.  COR.     The  lateral  area  of  a  right  circular  cone 
is  equal  to  one  half  the  product  of  the  slant  height  (or 
element}  and  the  circumference  of  the  base.    If  the  cone 
is  of  radius  r  and  slant  height  s,  the  lateral  area  =  irrs. 

167.  Theorem  XIX.     The  lateral  area  of  a  frustum  of 
a  regular  pyramid  is  equal  to  one  half  the  product  of 
the  slant  height  and  the  sum  of  the  perimeters  of  the 
bases.     What  kind  of  plane  figures  are  its  faces  ? 

168.  COR.  1.     The  lateral   area  of   a  frustum  of  a 
right  circular  cone  is  equal  to  one  half  the  product  of 
the  slant  height  and  the  sum  of  the  circumferences  of 
the  bases.     If  the  radii  of  the  bases  are  r1  and  rv  and  the 
slant  height  is  s,  the  lateral  area  is  7r«(r1  +  r2). 

169.  Midsections.     When  a  solid  has  parallel  bases,  the 
section  parallel  to  the  bases  and  half  way  between  them 
is  called  the  midsection. 

170.  COR.  2.     The  lateral  area  of  a  frustum,  of  a  right 
circular  cone  is  equal  to  the  product  of  the  slant  height 
and  the  circumference  of  its  midsection. 

If  the  radius  of  the  midsection  is  rm,  lateral «area  =  2  7rrms. 

171.  COR.  3.     The  lateral  area  of  a  frustum  of  aright 
circular  cone  is  equal  to  the  product  of  the  altitude  and 
the1  circumference  of  a  circle  whose  radius  is  the  per  pen- 


THE   PYRAMID   AND   THE   CONE 


299 


dicular  erected  at  the  midpoint  of  an  element,  and  termi- 
nated by  the  axis.     For  if  the  perpendicular  is  a  and  the 

altitude  A,  by  similar  triangles  -  =  — ,  and  ah  can  be  sub- 

">        rm 

stituted  for  srm  in  the  formula  of  §  170,  giving  2  irah. 

146.  Find  the  surface  of  a  hexagonal  tower  of  base  edge  10  ft. 
and  altitude  20  ft.  surmounted  by  a  pyramid  of  height  15  ft. 

147.  What  area  does  a  line  generate  if  it  is  revolved  around  a 
coplanar  axis,  its  projection  upon  the  axis  being  10  in.  and  the  length 
of  its  perpendicular  bisector  when  extended  to  the  axis  being  6  in.? 

172.  Prisms  Inscribed  in,  and  Circumscribed  about,  a  Pyr- 
amid. If  the  altitude  of  a  triangular  pyramid  is  divided 
into  equal  parts  by  planes  passed  parallel  to  the  base, 

1.  The  prisms  within  the  pyramid  having  the  sections 
formed  by  the  parallel  planes  as  bases,  and  each  having 
one  of  the  divisions  of  a  certain  edge  of  the  pyramid  as 
one  of  its  lateral  edges,  are  called  a  set  of  inscribed  prisms. 
In  Fig.  1,  U,  flf,  and  T  are  inscribed  prisms,  the  common 
edge  being  VC. 


FIG.  1. 


FIG.  2. 


2.  The  prisms  having  the  base  of  the  pyramid  and  the 
parallel  sections  as  bases,  and  each  having  one  of  the  divi- 
sions of  a  certain  edge  of  the  pyramid  as  one  of  its  lateral 
edges,  are  called  a  set  of  circumscribed  prisms.  In  Fig.  2, 
X,  r",  Z,  and  W  are  circumscribed  prisms,  the  common 
edge  being  VC, 


300         POLYHEDRONS,   CYLINDERS,   AND  CONES 

When  a  set  of  circumscribed  prisms  and  a  set  of  in- 
scribed prisms  are  formed  in  a  pyramid  by  the  use  of  the 
same  lateral  edge  and  the  same  parallel  planes,  they  are 
spoken  of  as  corresponding  sets  of  inscribed  and  circum- 
scribed prisms. 


*  173.    The  volume  of  a  triangular  pyramid  is  greater 
than  that  of  any  set  of  inscribed  prisms,  and  less  than 
that  of  any  set  of  circumscribed  prisms. 

*174.  In  any  triangular  pyramid,  the  difference  in 
volume  between  a  set  of  circumscribed  prisms  and  the 
corresponding  set  of  inscribed  prisms  is  the  circum- 
scribed prism  on  the  base  of  the  pyramid. 

As,  in  the  figures,  R  =  F,  S  =  Z,  T  =  TF,  therefore 
(x  +  r  +  z  +  w)  -  OR  +  8  +  r)  =  x. 

*  175.    If  the  number  of  equal  parts  into  which  the  alti- 
tude of  a  triangular  pyramid  is  divided  by  planes  par- 
allel to  the  base  is  increased  indefinitely,  the  volumes  of 
the  sets  of  inscribed  and  circumscribed  prisms  formed 
with  the  sections  as  bases  will  approach  the  volume  of 
the  pyramid  as  a  limit. 

For,  since  their  difference  is  a  prism  on  a  constant  base 
(ABC)  whose  altitude  can  be  made  indefinitely  small,  the 
difference  between  the  two  sets,  and  therefore  between 
either  and  the  pyramid,  can  be  made  to  approach  zero. 

176.  Theorem  XX.  Two  triangular  pyramids  having 
equivalent  bases  and  equal  altitudes  are  equivalent  in 
volume. 

Inscribe  sets  of  prisms,  using  the  same  number  of 
divisions  of  the  altitudes.  What  is  known  of  the  volumes 


THE  PYRAMID  AND  THE  CONE        301 

of  correspondingly  placed  prisms  in  the  two  pyramids  ? 
What  is  true  of  the  volumes  of  the  sets  of  prisms  ?  In- 
crease the  number  of  divisions  of  the  altitude.  Then  what 
is  true  of  the  sets  of  prisms  ?  Therefore  what  is  true  of 
the  pyramids  ?  Why  ? 

Alternative  Method.  The  two  pyramids  are  Cavalieri 
bodies. 

177.  Theorem  XXI.     A  triangular  prism  can  be  di- 
vided into  three  equivalent  triangular  pyramids. 

Pass  two  planes  through  vertices  of  the  prism  so  as  to 
divide  it  into  three  pyramids.  See  whether  they  are  equi- 
valent by  examining  their  bases  and  altitudes.  It  is  pos- 
sible to  prove  any  one  equivalent  to  any  other.  When  no 
easier  method  seems  possible,  choose  the  bases  of  the  two 
to  be  examined  so  that  they  are  in  the  same  plane. 

178.  COR.  1.     The  volume  of  a  triangular  pyramid 
equals  one  third  the  product  of  its  base  and  its  altitude. 

179.  COR.  2.      The   volume   of   any   pyramid    equals 
one  third  the  product  of  its  base  and  its  altitude. 

180.  COR.  3.     The  volume  of  a  circular  cone  equals  one 
third  the  product  of  its  base  and  its  altitude.     V  =  J  Trr^h. 

148.  Find  the  volume  of  a  right  circular  cone  of  radius  8  in. 
arid  altitude  15  in.,  and  of  the  regular  square  pyramid  circumscribed 
about  it. 

149.  If  square  pyramids  whose  lateral  faces  are  equilateral  triangles 
are  placed  on  all  the  faces  of  a  cube,  find  the  volume  of  the  resulting 
figure. 

150.  If  a  rectangular  parallelepiped  of  base  6  in.  by  8  in.,  and 
altitude  14  in.  is  hollowed  out  along  the  diagonal  planes  from  its  bases 
to  the  intersection  of  its  diagonals,  find  the  volume  of  the  remaining 
figure. 


302         POLYHEDRONS,   CYLINDERS,    AND   CONES 

151.  If  the  corners  are  cut  off  a  cube  by  passing  planes  through 
the  one-third  points  on  the  three  edges  from  each  vertex,  what  part 
of  its  volume  is  left  ? 

152.  Find  the  volume  of  the  solid  formed  by  passing  planes  through 
the  midpoints  of  the  three  edges  of  each  trihedral  angle  of  a  cube. 

181.  Prismatoids.  A  polyhedron  all  of  whose  vertices 
lie  in  two  parallel  planes  is  called  a  prismatoid.  Its  bases 
are  obviously  polygons  in  parallel  planes,  and  its  lateral 
faces  are  quadrilaterals  or  triangles. 

182  The  Prismatoid  Formula  and  its  Extension  to  Curved 
Surfaces. 

The  volume  of  any  prismatoid  is  expressed  by  the 
formula : 


where  h  is  the  altitude,  b±  and  b2  are  the  bases,  and  m  is 
the  midsection. 

Extension  of  the  Formula.  The  formula  applies  also  to 
solids  having  curved  surfaces,  if  (1)  their  bases  are  circles 
in  parallel  planes ;  (2)  their  generating  elements  are 
straight  lines  or  arcs  of  circles. 

This  extends  the  formula  to  cylinders  and  cones,  and  to 
certain  parts  of  spheres  that  have  not  yet  been  studied  ; 
it  also  applies  to  other  figures  with  which  elementary 
geometry  is  not  concerned.  ' 

The  proof  of  the  Prismatoid  Formula  is  not  given,  but 
for  the  case  of  polyhedrons  it  will  be  found  in  the  Appendix, 
§  320.  It  will  not  be  used  to  any  extent  to  prove  other 
formulas,  but  will  be  shown  to  agree  with  the  formulas 
found  for  the  various  figures  by  other  methods.  It  serves, 
then,  as  a  summary  for  almost  all  the  volume  formulas  of 


THE  PYRAMID  AND  THE  CONE        303 

solid  geometry,  and  might  well  be  used  for  all  the  volumes 
to  which  it  applies  were  it  not  for  the  fact  that  it  is  often 
harder  to  apply  than  the  particular  formula  for  the  figure 
in  question. 

For  example,  in  the  cone,  the  midsection  is  the  base 
divided  by  4  and  one  base  (considering  the  cone  a  pris- 
matoid)  is  zero,  so  the  prismatoid  formula  would  be 


which  agrees  with  the  regular  formula. 

183.  Theorem  XXII.  The  volume  of  the  frustum  of  a 
pyramid  equals  one  third  the  product  of  its  altitude  and 
the  sum  of  its  two  bases  and  the  mean  proportional  be- 
tween them. 


h  being  the  altitude,  and  b1  and  62  the  lower  and  upper 
bases,  the  one  farther  from  the  vertex  being  considered, 
for  convenience,  the  lower  base. 

FIRST  PROOF.  Continue  the  lateral  edges  through  the 
vertices  of  the  frustum  to  the  vertex  of  the  pyramid,  and 
consider  the  frustum  as  the  difference  of  two  pyramids. 
The  altitude  of  the  smaller  pyramid  can  be  obtained  from 

the   proportion    -L  =  -  —  ,      ,   where  x  is  the   unknown 
02  x* 

altitude.     Solve  this  for  x  by  taking  the  square  root  and 

applying  division,  finding  x  =  —       —  *-=•     Express  the 

V5j  -  V62 

volume  of  the  frustum  as  the  difference  of  two  pyramids 
in  terms  of  A,  #,  52,  bv  and  substitute  this  value  of  x. 


304         POLYHEDRONS,   CYLINDERS,   AND   CONES 

SECOND  PROOF.  Divide  a  triangular  frustum  into  three 
triangular  pyramids,  one  of  which  has  the  lower  base  of 
the  frustum  and  the  same  altitude,  a  second  of  which  has 
the  upper  base  of  the  frustum  and  the  same  altitude,  and 
prove  that  the  third  is  the  mean  proportional  between 
these  two.  In  order  to  do  this  take  the  ratio  of  the  third 
pyramid  to  each  of  the  others,  and  reduce  these  ratios  to 
simpler  form.  Use  the  same  method  as  in  Theorem  XXI, 
but  the  bases  of  the  pyramids,  instead  of  being  equal,  will 
have  equal  altitudes  only,  and  so,  in  turn,  will  be  pro- 
portional to  their  bases.  This  proof  must  be  extended  to 
any  frustum. 

THIRD  PROOF.  Use  the  Prismatoid  Formula,  express- 
ing the  midsection  in  terms  of  the  bases,  to  which  it  is  simi- 
lar. This  is  done  by  taking  the  ratio  of  each  of  the  bases 
to  the  midsection  and  adding  the  square  roots  of  the  equa- 
tions obtained. 

It  is  recommended  that  all  of  these  proofs  be  worked 
out,  for  they  are  all  valuable  applications  of  the  various 
methods  used. 

184.  COR.  The  volume  of  a  frustum  of  a  circular 
cone  is  equal  to  one  third  the  product  of  its  altitude  and 
the  sum  of  its  two  bases  and  the  mean  proportional  be- 
tween them. 


where  h  is  the  altitude,  and  ^  and  r2  are  the  radii  of  the 
bases. 

153.  A  monument  is  in  the  form  of  a  pillar  of  granite  1  ft.  in  di- 
ameter and  12  ft.  high,  surmounted  by  the  frustum  of  a  cone  1  ft. 
high  with  an  upper  base  of  radius  10  in.,  and  a  cone  of  height  6  in. 
How  much  does  the  stone  weigh  at  175  Ib.  to  the  cubic  foot? 


SUMMARY  OF   PROPOSITIONS  805 

185.  SUMMARY  OF  PROPOSITIONS 

I.   THK  STRAIGHT  LINE  : 

(1)  Sects  equal: 

The  edges  of  a  prism  are  equal  (107). 
The  elements  of  a  cylinder  are  equal  (107). 

(2)  Sects  proportional : 

The  edges  of  a  pyramidal  space  cut  by  parallel  planes  are 
proportional  (149). 

II.   TANGENT  PLANES  AND  SECTIONS  : 

(1)  Tangent  planes : 

The  plane  determined  by  an  element  and  a  tangent  to  the 
base  of  a  cylinder  (or  a  cone)  is  tangent  to  the  cylinder 
(or  the  cone),  and  conversely  (119,  120,  163,  164). 

(2)  Sections: 

(a)  of  a  prismatic  space : 

Parallel  sections  are  congruent  polygons  (108). 

Any  pair  of  opposite  faces  of  a  parallelepiped  may  be 

considered  its  bases  (126) ;    its  opposite  faces  are 

congruent  (126). 

(b)  of  a  cylindrical  space  : 

Parallel  sections  are  congruent  (109). 
A   section   through   an   element  of  a  cylinder   is  a 
parallelogram  (117). 

(c)  of  a  pyramidal  space  : 

Parallel  sections  are  similar  polygons  (150)  ;    their 
areas  are  proportional  to  the  squares  of  their  dis- 
tances from  the  vertex  (150). 
(e?)  of  a  cone : 

A  section  parallel  to  the  base  of  a  circular  cone  is  a 
circle  (161). 

A  section  through  an  element  is  a  triangle  (160). 

III.   LIMITING  CASES: 

If  the  number  of  faces  of  a  prism  (or  a  pyramid)  of  regular  base, 
inscribed  in  or  circumscribed  about  a  circular  cylinder  (or  a 
cone),  is  increased  indefinitely,  the  lateral  area  and  the  volume 


306         POLYHEDRONS,   CYLINDERS,   AND   CONES 

of  the  prism  (or  the  pyramid)  approaches  as  its  limit  the  lateral 
area  and  the  volume  of  the  cylinder  (or  the  cone)  (116,  159). 
If  the  number  of  equal  parts  into  which  the  altitude  of  a  trian- 
gular pyramid  is  divided  by  planes  parallel  to  the  base  is  in- 
creased indefinitely,  the  volumes  of  the  sets  of  inscribed  and 
circumscribed  prisms  formed  with  the  sections  as  bases  ap- 
proach the  volume  of  the  pyramid  as  a  limit  (175). 

IV.   LATERAL  AREA  : 

(1)  of  a  prism  =  edge  x  perimeter  of  a  right  section  (121); 

(2)  of  a  right  circular  cylinder  =  altitude  x  perimeter  of  the 

base  =  27rrA,  where  h  is  the  altitude,  r  is  the  radius  (122); 

(3)  of  a  regular  pyramid  =  ^  slant  height  x  perimeter  of  the 

base  (165) ; 

(4)  of  a  right  circular  cone  =  |  slant  height  x  perimeter  of  the 

base  =  TITS,  where  r  is  the  radius,  s  is  the  slant  height 
(166); 

(5)  of  a  frustum  of  a  regular  pyramid  =  |  slant  height  x  the 

sum  of  the  perimeters  of  the  bases  (167); 

(6)  of  a  frustum  of  a  right  circular  cone  =  |  slant  height  x  the 

sum  of  the  perimeters  of  the  bases  =  ITS  (ri  +  r2),  where 
r\  and  rz  are  the  radii,  —  2?rsrw,  where  rm  is  the  radius  of 
the  midsection,  —  2  irdh,  where  a  =  the  perpendicular  bi- 
sector of  an  element  extended  to  the  axis  (168,  170,  171). 

V.  VOLUMES  : 

(1)  Congruent : 

Right  prisms  or  cylinders  having  congruent  bases  and  equal 
altitudes  are  congruent  (123). 

(2)  Equivalent : 

Two  Cavalieri  bodies  are  equivalent  (137). 

Prisms  cut  from  the  same  prismatic  space  and  having  equal 

lateral  edges  are  equivalent  (138). 
An  oblique   parallelepiped  is   equivalent  to  a  rectangular 

parallelepiped  having  an  equivalent  base  and  an  equal 

altitude  (139). 
The  parts  of   a   parallelepiped  made   by  passing   a  plane 

through  two  opposite  edges  are  equivalent  (141). 


SUMMARY   OF  PROPOSITIONS  307 

Triangular  pyramids  of  equivalent  bases  and  equal  altitudes 
are  equivalent  (176). 

A  triangular  prism  is  equivalent  to  three  equivalent  pyra- 
mids (177). 

(3)  Proportional: 

Rectangular  parallelepipeds  having  two  dimensions  equal 

are  to  each  other  as  the  third  dimension  (132). 
Rectangular  parallelepipeds  having  one  dimension  equal  are 

to  each  other  as  the  products  of  the  other  two  dimensions 

(133). 
Rectangular  parallelepipeds  having  no  dimensions  equal  are 

to  each  other  as  the  products  of  all  three  dimensions  (134). 

(4)  The  Pyramid  and  its  inscribed  and  circumscribed  prisms : 

A  triangular  pyramid  is  less  than  the  circumscribed,  and 
greater  than  the  inscribed  prisms  (173). 

The  difference  between  the  sets  of  circumscribed  and  in- 
scribed prisms  is  the  circumscribed  prism  on  the  base  of 
the  pyramid  (174). 

(5)  Formulas  for  volume  : 

(a)  of  a  rectangular  parallelepiped  =  the  product  of  its  di- 
mensions =  base  x  altitude  (136); 

(6)  of  any  parallelepiped  =  base  x  altitude  (140); 

(c)  of  a  prism  =  base  x  altitude  (142,  143); 

(</)  of  a  circular  cylinder  =  base  x  altitude  (144); 

(e)  of  a  triangular  pyramid  =  }  base  x  altitude  (178); 

(/)  of  any  pyramid  =  £  base  x  altitude  (179); 

(g)  of  a  circular  cone  =  £  base  x  altitude  (180); 

(A)  of  a  prismatoid  =  £  altitude  x  the  sum  of  the  bases  and 
four  times  the  midsection  (182); 

(£)  of  a  frustum  of  a  pyramid  =  j  altitude  x  the  sum  of  the 
bases  and  the  mean  proportional  between  them  (183); 

(/)  of  a  frustum  of  a  circular  cone=~  x  altitude  x  the  sum 
of  the  squares  of  the  radii  and  their  product,  which 

-sTS.ffp  4.  r22  _j.  rir2)  where  h  is  the  altitude,  and  rt 

3 
and  r2  are  the  radii  of  the  bases  (184). 


308         POLYHEDRONS,   CYLINDERS,   AND   CONES 

186  ORAL  AND  REVIEW  QUESTIONS 

Explain  why  the  circular  cylinder  and  cone  can  be  assumed  to  be 
the  limits  of  the  inscribed  or  the  circumscribed  prism  and  pyramid  with 
regular  bases.  How  is  this  fact  used?  What  changes  are  made  in 
the  formulas  when  they  are  applied  to  the  circular  figures?  What 
application  of  similar  figures  occurs  in  this  book  ?  If  the  altitude  of 
a  pyramid  is  divided  into  ten  equal  parts  by  planes  passed  parallel  to 
the  base,  what  ratio  has  the  area  of  each  of  the  sections  to  the  area  of 
the  base?  State  the  theorem  used,  and  explain  briefly  why  it  is 
true.  State  formulas  for  the  lateral  areas  of  prism,  right  prism, 
parallelepiped,  right  parallelepiped,  rectangular  parallelepiped,  cir- 
cular cylinder,  right  circular  cylinder,  right  circular  cone,  regular 
pyramid,  frustum  of  a  regular  pyramid,  frustum  of  a  right  circular 
cone.  Why  must  it  be  a  regular  pyramid?  why  a  right  circular  cone? 
Define  a  plane  tangent  to  a  cylinder  or  a  cone.  How  is  it  determined  ? 
Explain  through  what  succession  of  propositions  the  volume  of  a 
rectangular  parallelepiped  is  found.  Follow  the  steps  from  the 
volume  of  a  rectangular  parallelepiped  to  that  of  any  pyramid.  Does 
the  pyramid  need  to  be  regular  for  this  formula  to  hold?  Does  the 
cone  need  to  be  circular  for  its  volume  formula  to  be  true  ?  Explain 
in  full.  State  Cavalieri's  Theorem.  What  facts  need  to  be  established 
about  two  figures  in  order  that  this  theorem  can  be  used  ?  When  does 
the  Prismatoid  Formula  hold  ?  State  the  volume  formulas  for  all  the 
solids  studied  in  this  book.  What  is  known  of  parallel  sections  of  a 
prism?  of  a  cylinder?  of  a  pyramid?  of  a  cone?  of  a  section  through 
an  edge  of  a  prism?  through  an  edge  of  a  pyramid?  through  an 
element  of  a  cylinder?  through  an  element  of  a  cone?  through  two 
opposite  edges  of  a  parallelepiped  ?  cutting  two  pairs  of  its  opposite 
faces  ?  through  three  concurrent  edges  ?  What  sections  are  congruent  ? 

GENERAL  EXERCISES 

154.  If  three  planes  are  perpendicular  to  each  other,  the  square  of 
the  distance  from  any  point  to  their  common  vertex  equals  the  sum 
of  the  squares  of  the  distances  from  that  point  to  the  three  planes. 

155.  If  at  the  top  of  a  chimney  whose  flue  is  8  in.  by  10  in.  a 
round  pipe  is  to  be  placed,  how  large  should  it  be  to  give  the  same 
opening  ?     If  two  equal  round  chimneys  are  to  be  placed  side  by  side, 
how  large  should  each  be  to  give  the  same  total  opening  ? 


GENERAL   EXERCISES  309 

156.  What  kind  of  surface  is  generated  by  opening  a  door? 

157.  If  in  the  base  of  a  right  circular   cylinder  of  radius  r  and 
altitude  A,  two  diameters  are  drawn  perpendicular  to  each  other,  and 
four  planes  perpendicular'  to  the  base  are  passed  through  the  ends  of 
each  two  consecutive  radii  formed  by  these  diameters,  what  part  of 
the  volume  is  inclosed  by  these  planes? 

158.  If  a  right  section  of  a  certain  hexagonal  pillar  of  diameter 
20  ft.  is  taken,  it  will  show  that  each  surface  is  curved  toward  the 
center  along  a  90°  arc,  all  the  arcs  being  equal.     The  pillar  is  100  ft. 
high,  the  last  20  ft.  being  pyramidal,  with  the  base  a  right  section  of 
the  lower  part  of  the  pillar.     If  this  pillar  stands  on  a  frustum  of  a 
regular  pyramid,  which  has  edges  30  ft.  and  20  ft.,  and  the  altitude 
15  ft.,   how  many  cubic  yards  of  material  were  used  in  the  entire 
construction? 

159.  In  some  hospitals,  floors  are  joined  to  the  walls  by  a  curved 
surface  instead  of  by  an  edge.     If  the  concrete  is  shaped  by  the  use  of 
a  trowel  in  the  form  of  the  surface  of  a  cylinder  of  radius  2  in.,  used 
with  its  convex  surface  toward  the  edge,  find  how  much  concrete  to 
the  linear  foot  would  be  needed  to  change  an  edge  into  the  curved 
surface. 

160.  If  concrete  blocks   are  made   in   the  form  of  a  rectangular 
parallelepiped   6   in.  x  8   in.  x  14   in.,   and   they   are    hollowed    out, 
parallel  to  the  edges  of  length  8  in.,  in  the  form  of  a  parallelepiped 
with  cylindrical  ends,  the  entire  length  of  the  hollow  being  10  in., 
what  part  of  a  wall  built  of  these  blocks  will  be  air  space? 

161.  Which  holds  more,  a  suit  case  14  in.  x  2  ft.  x  6  in.,  or  one 
13  in.  x  23  in.  x  7  in.  ? 

162.  How  much  air  space  is  gained  in  a  house  20  ft.  wide,  with 
stories  8^  ft.  and  8  ft.  high  respectively,  by  adding  two  semicircular 
bay  windows  across  the  front,  allowing  1  ft.  for  the  thickness  of  the 
middle  partition,  and  six  inches  for  each  side  wall  ? 

163.  The  sign  on  a  corner  store  extends  10  ft.  along  one  street, 
with  a  quarter  circle  at  the  corner,  and  a  17-ft.  extension  on  the  other 
street.     If  the  sign  is  2  ft.  wide  and  its  area  is  63.4248  sq.  ft.,  what 
is  the  radius  of  the  corner  ? 

164-    Given  a  cone  of  radius  r  and  height  h,  find  what  fraction  of 
its  volume  is  lost  by  cutting  out  a  pyramid  with  its  base  a  regular 


310         POLYHEDRONS,   CYLINDERS,   AND   CONES 

polygon  inscribed  in  the  base  of  the  cone,  if  the  polygon  has  (1)  3 
sides,  (2)  4  sides,  (3)  6  sides,  (4)  8  sides. 

165.  A  shed  is  built  9  ft.  high  in  the  front  and  7  ft.  high  in  the 
back,  with  a  floor  12  ft.  across  and  10  ft.  from  front  to  back.  What 
is  the  air  space,  the  amount  of  lumber  in  the  sides,  and  the  number 
of  square  yards  of  roofing  needed  to  cover  it,  if  the  roof  extends  1  ft. 
all  around,  and  10  %  is  allowed  for  laps  and  waste? 

\166.   If  the  diagonal  of  a  cube  is  4V3  in.,  find  its  volume. 

167.  In  a  right  circular  cone,  of  lateral  area  I  and  base  b,  find  the 
altitude. 

168.  If  the  lateral  area  of  a  right  circular  cone  is  equivalent  to 
the  area  of  the  base,  express  the  volume  in  terms  of  the  radius. 

169.  It  has  been  found  that  in  irrigating  by  concrete  ditches  whose 
sides  are  perpendicular  to   the   bottom,  there   is   the   least  possible 
friction  for  a  given,  volume  of  water  when  the  depth  of  the  ditch  is 
half  its  width.     As  the  friction  depends  on  the  amount  of  bottom  and 
side  wall,  such  a  ditch  is  also  most  economical  in  materials.     Find 
the  most  economical  rectangular  ditch  to  carry  56,320  cu.  yd.  of  water 
to  the  mile,  and  show  that  one  carrying  the  same  amount  of  water, 
but  having  either  higher  or  lower  side  walls,  would  be  less  economical. 
(Do  this  by  finding  the  total  area  for  each  of  three 'cases.) 

170.  If  a  ditch  is  to  have  its  side  walls  at  a  45°  slope,  it  is  most 
economical  when  the  width  at  the  bottom    is  approximately  four 
fifths  of  the  depth.     Find  the  contents  of  such  a  ditch  5  ft.  deep  and 
a  hundred  yards  long. 

171.  It  has  been  found  that  the  most  economical  cylindrical  can 
to  use  for  canning  fruit  or  vegetables  is  one  whose  altitude  equals  its 
diameter.     Find  the  volume  of  such  a  can  3  in.  in  height,  and  show 
that  one  of  the  same  volume,  but  of  slightly  greater  height  would 
require  more  tin  to  inclose  it. 

*  172.  If  a  cylindrical  shell  of  external  radius  6  in.  and  thickness  £ 
in.,  has  one  conical  end,  the  slant  height  of  the  cone  being  12  in. 
and  the  total  length  of  the  cylindrical  part  of  the  shell  being  24£  in., 
find  the  weight  of  the  metal  used,  at  450  Ib.  to  the  cubic  foot.  (Note 
that  the  difference  in  the  altitude  of  the  inside  and  outside  cones  is 
not  \  in.) 


GENERAL   EXERCISES  311 

173.  A  rectangular  building  stone  6  in.  by  12  in.  by  3  ft.  is  dropped 
into  a  cistern  of  radius  3  ft.     How  much  does  it  raise  the  water  ? 

174.  A  body  of  irregular  shape  was  placed  in  a  cylindrical  vessel 
of  diameter  6  in.  partly  filled  with  water.     What  was  its  volume  if  it 
raised  the  water  1|  in.? 

175.  What  would  be  the  smallest  amount  of  tin  that  would  make 
a  cylindrical  can  to  contain  one  quart,  counting  57|  cu.  in.  to  the 
quart,  and  allowing  5%  waste  for  cutting  and  lapping? 

176.  The  greatest  right  circular  cylinder  that  can  be  inscribed  in 
a  right  circular  cone  is  one  whose  altitude  is  one  third  that  of  the 
cone.     Show  the  ratio  of  its  volume  to  that  of  the  cone,  and  the  ratio 
of  its  area  to  that  of  the  cone. 

177.  How  much  water  will  a  pail  hold  if  it  is  16  in.  deep,  a  foot 
in  diameter  at  the  bottom,  and  14  in.  in  diameter  at  the  top? 

178.  Find  the  volume  of  a  polyhedron  having  one  base  a  rectangle 
6  ft.  by  4  ft.,  the  other  a  square  of  edge  2  ft.,  with  each  side  parallel 
to  the  corresponding  side  of  the  rectangle,  and  an  altitude  of  8  ft. 

179.  A  hole  is  dug  in  the  form  of  the  frustum  of  a  regular  square 
pyramid  of  upper  edge  4  ft.,  lower  edge  2  ft.,  depth  4|  ft.     Find  the 
amount  of  earth  removed.     If  the  rain  fills  it  two  thirds  of  the  way 
up,  what  part  of  its  volume  is  still  above  water? 

180.  How  much   tin  is  required  to  make  a  funnel  in   the  form 
of  the  frustum  of  a  cone  5  in.  high,  with  radii  3  in.  and  \  in.,  with  a 
cylinder  2  in.  long  on  the  smaller  end,  allowing  10  %  for  waste  ? 

181.  What  is  the  cubic  content  of  a  concrete  support  in  the  form 
of  a  regular  frustum  with  square  bases  of  edges  2  ft.  and  18  in.,  and 
height  18  in.? 

182.  A  street  car  has  side  walls  6  ft.  high,  above  each  of  which  is 
a  quarter  circle  of  radius  1£  ft.,  the  center  being  inside  of,  and  on 
the  level  with,  the  top  of  the  side  wall,  and  at  the  end  of  this  arc,  a 
second  side  wall  of  height  15  in.     If  the  total  width  of  the  car  is 
8  ft.,  and  its  length  is  50  ft.,  how  many  cubic  feet  of  air  are  there  to 
a  passenger  when  one  hundred  and  ten  passengers  are  in  the  car? 

183.  Measure  the  room  in  which  your  class  recites,  and  calculate 
the  number  of  cubic  feet  of  air  space  per  person. 

184.  A  suburban  railroad,  in  changing  the  position  of  its  tracks 
to  avoid  grade  crossings,  inclosed  its  right  of  way  by  concrete  walls 


312         POLYHEDRONS,   CYLINDERS,   AND   CONES 

12  ft.  high,  1  ft.  wide  at  the  top,  and  3  ft.  wide  at  the  bottom,  one 
face  being  perpendicular  to  the  plane  of  the  tracks.  If  12  mi.  of 
track  were  inclosed,  how  many  cubic  yards  of  concrete  were  built  ? 

185.  How  many  cubic  yards  of  concrete  would  be  needed  to  taper 
off  the  wall  described  in  Ex.  184  at  a  45°  angle  at  the  level  ? 

The  Wedge.  A  solid  whose  base  is  a  rectangle,  and  two  of  whose 
four  lateral  faces  meet  in  an  edge  parallel  to  the  base,  is  called  a 
wedge.  The  edge  parallel  to  the  base  is  called  the  edge  of  the  wedge. 

186.  A  wedge  has  a  base  2  in.  by  6  in.,  the  altitude  12  in.,  and  an 
edge  4  in.     Find  its  volume. 

187.  Find  the  volume  of  a  right  circular  cone  whose  volume  equals 
its  lateral  surface  times  one  sixth  the  radius  of  the  base,  which  is 
\/3  in.  long. 

^  188.  A  cube  is  hollowed  out  from  one  face,  in  the  form  of  a  square 
pyramid  whose  lateral  faces  are  equilateral  triangles.  What  part  of 
the  cube  is  left  ?  What  is  the  total  area  of  the  figure  if  the  edge  is 
10  in.? 

189.  Find  the  amount  of  space  between  the  surface  of  a  regular 
"^hexagonal  pyramid  of  base  edge  1  ft.,  lateral  edge  2  ft.,  and  that  of 

the  inscribed  cone. 

190.  If  the  total  area  of  a  cylinder  of  radius  r  is  a,  and  its  volume 
is  br,  express  r  in  terms  of  a  and  b. 

191.  A  subway  stair  has  a  platform  6  ft.  by  9  ft.,  approached  on 
opposite  sides  by  two  flights  of  stairs  of  10  steps  each,  each  flight 
being  6  ft.  wide,  and  each  step  having  a  rise  of  7  in.,  and  a  tread  of 
9  in.     If  the  concrete  is  2  in.  thick,  how  much  air  space  is  there  in 
the  closet  beneath  the  platform  and  the  stairs? 

192.  A  steam  shovel  has  a  scoop  4  ft.  deep,  3  ft.  wide,  and  4|  ft. 
long,  the  cross  section  being  a  square  with  a  semicircle  on  one  end. 
How  many  times  would  the  scoop  have  to  be  filled  to  dig  a  cellar  30  ft. 
wide,  45  ft.  long,  and  6  ft.  deep,  with  two  bow  windows,  one  semi- 
circular, of  diameter  10  ft.,  the  other  a  three-quarter  circle  of  radius 
6  ft.,  around  one  corner  of  the  house  as  a  center,  if,  on  the  average, 
the  scoop  is  two  thirds  full  each  time? 

193.  How  many  cubic  fest  of  masonry  are  there  in  a  round  chimney 
50  ft-  tall,  8  ft.  in  external,  and  5  ft.  in  internal,  diameter  at  the  bottom, 
and  5  ft.  in  external,  and  4  ft.  in  internal,  diameter  at  the  top? 


GENERAL   EXERCISES  313 

194-  How  much  cloth  is  there  in  a  skirt  24  in.  around  the  waist, 
6  ft.  around  the  bottom,  and  42  in.  long,  allowing  10%  for  seams  and 
fi  tting  V 

195.  To  cut  a  cube  by  a  plane  so  that  the  section  shall  be  a  regular 
hexagon. 

196.  A  measure  is  in  the  form  of  a  right  circular  frustum  of  radii 
2  in.  and   3   in.     If   it   holds  two  quarts,  find   its    height.     If  it  is 
graduated  to  pints,  at  what  points  on  an  element  should  the  gradua- 
tions be  made? 

197.  What  would  be  the  most  economical  size  for  a  factory  fire 
tank  to  hold  2000  gal.  of  water? 

198.  Pencils  T\  in.  in  diameter  have  their  halves  cut  from  wood 
/a  in.  thick.     If  the  lead  is  ^  in.  in  diameter,  what  per  cent  of  the 
wood  is  wasted  ? 

199.  How  much  earth  is  removed  in  digging  12  post  holes  18  in.  in 
diameter  and  5  ft.  deep? 

j-200.   The  midpoints  of  two  pairs  of  opposite  edges  of  a  tetrahedron 
are  coplanar. 

201.  A  section  of  a  tetrahedron  by  a  plane  parallel  to  two  opposite 
edges  is  a  parallelogram. 

202.  The  lines  joining  the  vertices  of  a  tetrahedron  to  the  centroids 
of  the  opposite  faces  meet  in  the  three-fourths  point  of  each. 

203.  To  pass  a  plane  through  a  given  point  tangent  to  a  given 
circular  cone. 

204-  To  pass  a  plane  through  a  given  point  tangent  to  a  given 
circular  cylinder. 

205.  If  the  number  of  square  units  in  the  area  of  a  cube  equals  the 
number  of  cubic  units  in  its  volume,  find  its  area*. 

206.  A  watering  trough  was  built  by  nailing  two  boards  together  at 
right  angles,  one  against  the  edge  of  the  other,  and  closing  the  ends. 
If  the  boards  were  1  in.  thick,  and  the  one  nailed  against  the  edge 
of  the  other  was  the  wider  by  1  in.,  find  the  amount  of  water  such 
a  trough  1  ft.  5  in.  wide  across  the  open  top  and  8  ft.  long  would 
hold. 

207.  If  the  radius  of  one  cylinder  equals  the  altitude  of  a  second 
and  vice  <v»r.s-a,  what  is  the  ratio  of  their  volumes? 


BOOK   VIII.     POLYHEDRAL   ANGLES 
AND   THE   SPHERE 

SECTION  I.     DEFINITIONS;   SECANTS  AND  TANGENTS 

187.  The  Sphere.     The  geometric  solid  bounded  by  a 
surface  all  points  of  which  are  equidistant  from  a  point 
within,  called  the  center,  is  a  sphere.     The  bounding  sur- 
face is  called  a  spherical  surface. 

188.  Generation  of  a  Sphere.     A  sphere  can  be  gener- 
ated by  the  rotation  of  a  semicircle  about  its  diameter  as 
an  axis,  for  all  points  on  the  surface  of  the  solid  thus  gen- 
erated will  be  at  a  distance  from  the  center  of  the  semi- 
circle equal  to  a  radius. 

189.  Radii  and  Diameters.     A  line  from  the  center  of  a 
sphere  to  its  surface  is  a  radius,  while  a  line  through  the 
center,  joining  two  points  on  the  surface,  is  a  diameter. 

*  190.    All  radii  of  a  sphere  are  equal.    All  diameters 
of  a  sphere  are  equal. 

*191.    Spheres  having  equal  radii  are  congruent  and 
conversely. 

For,  if  superposed  with  centers  coinciding  (that  is, 
placed  so  that  they  are  concentric),  no  point  on  one  sur- 
face could  be  off  the  other  surface.  (Why  ?)  Note  also 
that  equivalent  spheres  must  have  equal  radii,  for  if  not, 
the  one  having  the  longer  radius  could  contain  the  other. 

314 


DEFINITIONS  315 

From  this  it  follows  that  only  the  word  equal  need  be  used 
for  spheres,  for  with  them  there  is  no  distinction  between 
equivalence  and  congruence.  Compare  with  the  circle. 

*  192.    The  locus  of  points  at  a  given  distance  from  a 
given  point  is  the  surface  of  a  sphere  having  the  given 
point  as  its  center,  and  the  given  distance  as  its  radius. 

208.   Find  the  locus  of  the  vertex  of  the  right  angle  of  a  right 
triangle,  having  a  given  fixed  hypotenuse. 

193.    Relative   Positions  of  a  Point  and  a  Sphere.     A 

point  is  within  a  sphere  if  its  distance  from  the  center  is 
less  than  the  length  of  the  radius;  on  the  surface,  if  its  dis- 
tance from  the  center  is  equal  to  the  length  of  the  radius; 
outside,  if  its  distance  from  the  center  is  greater  than  the 
length  of  the  radius.  This  follows  from  the  definition  of 
a  sphere,  and  the  ordinary  meanings  of  "within"  and 
"outside." 

*  194.    One,  and  but  one,  spherical  surface  can  be  drawn 
through  four  non-coplanar  points.     For  there  is  one  and 
but  one  point  equidistant  from  the  four  points.     See  §  91. 

195.  Inscribed  Polyhedrons.     A  sphere  is  said  to  be  cir- 
cumscribed about  a  polyhedron  if  its  surface  contains  the 
vertices  of  the  polyhedron;  the  polyhedron  is  said  to  be 
inscribed  in  the  sphere.     §194  might  have  been  worded: 
One  and  but  one  sphere  can  be  circumscribed  about  a  given 
tetrahedron. 

196.  Relative  Positions  of  a  Line  and  a  Sphere.     A  line 
can  lie  wholly  outside  a  sphere,  for  the  sphere  is  limited 
in  size,  or  it  can  have  one  or  more  points  in  common  with 
the  sphere.     If  the  distance  of  the  line  from  the  center  is 
equal  to  a  radius,  it  has  but  one  point  in  common  with 
the    sphere    (why  ?),    and    so   is    called    tangent     to    the 


316         POLYHEDRAL   ANGLES   AND   THE   SPHERE 

sphere,  the  point  being  called  the  point  of  contact.  If  the 
distance  of  the  line  from  the  center  is  less  than  a  radius, 
two,  and  but  two,  lines  of  length  equal  to  a  radius  can  be 
drawn  to  it  from  the  center  (determining  the  plane  of  the 
line  and  the  center  and  using  plane  geometry) ;  so  the  line 
meets  the  surface  in  two  points  and  is  called  a  secant. 
Show  that  a  line  tangent  to  a  sphere  is  not  tangent  to 
all  the  circles  through  its  point  of  contact. 

209.  A  line  whose  distance  from  the  center  of  a  sphere  is  more 
than  the  length  of  a  radius  does  not  meet  the  surface  of  the  sphere. 

210.  A  line  tangent  to  a  sphere  must  be  perpendicular  to  the  ra- 
dius drawn  to  its  point  of  contact. 

211.  The  plane  perpendicular  to  a  tangent  at  its  point  of  contact 
passes  through  the  center  of  the  sphere. 

197.    Relative  Positions  of  a  Plane  and  a  Sphere.     A 

plane  is  outside  a  sphere  if  its  distance  from  its  center  is 
more  than  the  length  of  a  radius.  Why  ? 

A  plane  is  tangent  to  a  sphere  if  it  has  one  point  in  com- 
mon with  the  sphere.  The  possibility  of  such  a  plane  is 
shown  in  §  198.  The  common  point  is  called,  as  in  the 
case  of  the  tangent  line,  the  point  of  contact. 

A  plane  is  a  secant  plane  if  it  cuts  the  sphere,  the  inter- 
section with  the  surface  necessarily  (§  101)  being  a 
closed  line.  See  §  203. 

*  198.    A  plane  perpendicular  to  a  radius  of  a  sphere 
at  its  surface  extremity  is  tangent  to  the  sphere. 

This  proves  that  there  can  be  a  plane  tangent  to  a 
sphere  at  any  point  on  its  surface.  Why  ? 

*  199.    A  plane  tangent  to  a  sphere  is  perpendicular  to 
the  radius  drawn  to  the  point  of  contact. 


SECANTS   AND   TANGENTS  317 

For  the  point  of  tangency  is  nearest  the  center.  This 
proves  that  there  can  be  but  one  tangent  plane  at  a  point 
on  the  surface.  Why?  §§  198  and  199  show  that  a  tan- 
gent plane  is  a  plane  whose  distance  from  the  center 
equals  the  length  of  the  radius.  Similarly  a  secant  plane 
is  one  whose  distance  from  the  center  is  less  than  the 
length  of  the  radius. 

*  200.    If  a  plane  is  tangent  to  a  sphere,  a  perpendic- 
ular to  it  at   the  point  of  contact  passes  through  the 
center  of  the  sphere. 

212.  All  lines  tangent  to  a  sphere  at  a  point  are  perpendicular  to 
the  radius  drawn  to  that  point. 

213.  All  lines  tangent  to  a  sphere  at  a  point  are  in  the  plane  tan- 
gent to  the  sphere  at  that  point. 

214>  Two  lines  tangent  to  a  sphere  at  a  point  determine  the  plane 
tangent  to  the  sphere  at  that  point. 

201.  Circumscribed  Polyhedrons.  A  polyhedron  is  said 
to  be  circumscribed  about  a  sphere,  and  the  sphere  to  be 
inscribed  in  the  polyhedron,  if  the  sphere  is  tangent  to  all 
the  faces  of  the  polyhedron. 

*  202.    One,   and  but  one,   sphere  can  be  inscribed   in 
any  given  tetrahedron.       For    there    is    one    point   equi- 
distant from  its  faces. 

203.  Theorem  I.  Every  plane  section  of  a  sphere  is  a 
circle  whose  center  is  the  foot  of  the  perpendicular 
from  the  center  of  the  sphere  to  its  plane. 

Plane  sections  of  a  sphere  are  called  circles  of  the  sphere. 
"Circle  "  is  usually  used  for  "circle  of  a  sphere." 

215.  If  a  secant  plane  is  gradually  moved  farther  from  the  center 
of  a  sphere,  describe  the  change  in  its  intersection  with  the  surface. 


318 


POLYHEDRAL  ANGLES   AND   THE   SPHERE 


216.   A  line  tangent  to  a  sphere  is  tangent  to  every  circle  through 
the  point  of  contact,  in  whose  plane  it  lies. 

204.  Axis  and  Poles  of  a  Circle.     The  diameter  perpen- 
dicular to  a  circle  is  called  its  axis ;  the  points  where  the 
axis  cuts  the  surface  of  the  sphere  are  called  the  poles  of 
the  circle. 

205.  COR.  1.     A  line  perpendicular  to  a  circle  at  its 
center,  or  one  joining  the  center  of  the  circle  to  the  center 
of  the  sphere  (unless  they  are  the  same  point},  is  the 
axis  of  the  circle. 

206.  COR.  2.     Sections  through  the  center  of  a  splwre 
are  all  equal,  and  are  the  largest  circles  of  the  spJiere. 
Of  others,  two  that  are  equidistant  from  the  center  are 
equal,  and  conversely ;  and,  of  two  not  equidistant  from 
the  center,  the  nearer  is  the  greater,  and  conversely. 

207.  Great  and  Small  Circles.     A  section  through  the 
center  of  a  sphere  is  called  a  great  circle ;  any  other  sec- 


tion is  called  a  small  circle.  In  the  diagram,  G  is  a  great 
circle,  and  8  is  a  small  circle.  The  axis  of  S  is  PPf,  its 
poles  being  P  and  P'.  What  relation  to  one  another  have 
r  (the  radius  of  the  sphere),  rf  (the  radius  of  the  small 
circle),  and  d  (its  distance  from  the  center)? 


SECANTS  AND   TANGENTS  319 

The  circles  of  latitude  and  the  meridians  on  the  earth's 
surface  are  familiar  examples  of  circumferences  of  circles 
of  a  sphere.  The  equator  and  the  meridians  are  circum- 
ferences of  great  circles. 

*208.  The  center  of  a  great  circle  is  the  center  of  the 
sphere. 

*  209.   Any  two  great  circles  of  a  sphere  intersect  in  a 
diameter. 

*  210.   A  great  circle  bisects  the  sphere  and  its  surface. 

*211.  Two  points  on  the  surface  (not  extremities  of  a 
diameter}  and  the  center  of  the  sphere  determine  a  great 
circle ;  any  three  points  on  the  surface  determine  a  circle. 

212.  COR.   3.     Parallel   circles  of  a  sphere  have  the 
same  axis  and  the  same  poles. 

The  circles  of  latitude  on  the  earth's  surface  have  the 
axis  of  the  earth  as  their  axis,  and  its  poles  as  their  poles. 

213.  COR.  4.     A  great  circle  through  the  poles  of  an- 
other circle  is  perpendicular  to  it ;  a  great  circle  perpen- 
dicular to  another  circle  contains  its  axis  and  its  poles. 
This  is  sometimes  stated  :  For  a  great   circle  to  be  per- 
pendicular to  another  circle,  it   is  necessary,  and  it  is 
sufficient,  that  it  contain  its  poles. 

217.  A  great  circle  that  contains  one  of  the  poles  of  another  circle 
must  contain  the  other  pole  also. 

214.  COR.  5.     The  locus  of  points  in  a  sphere  equidis- 
tant from  all  points  on  the  circumference  of  a  circle  is 
the  axis  of  the  circle ;  the  poles  of  a  circle  are  equidistant 
from  all  points  on  its  circumference. 


320         POLYHEDRAL   ANGLES   AND   THE   SPHERE 

215.  Polar  Chords  and  Polar  Distances.  The  chords 
from  the  pole  of  a  circle  to  points  on  its  circumference 
are  called  polar  chords;  the  great  circle  arcs  from  its 
pole  to  points  on  its  circumference  are  called  polar  dis- 
tances. Unless  otherwise  stated  the  chords  and  distances 
from  the  nearer  pole  are  meant. 

*  216.    The  polar  distances  of  the  same,  or  of  equal,  cir- 
cles on  a  sphere  are  equal. 

217.  A  Quadrant.     One  quarter  of   the   circumference 
of  a  great  circle  is  called  a  quadrant  of  the  sphere.     The 
degree  measure  of  a  quadrant  is  90°. 

*  218.    The  polar  distance  of  a  great  circle  is  a  quad- 
rant, and  conversely. 

Use  the  central  angles  subtended  by  the  polar  distances. 

*  219.    If  two  points  on  a  spherical  surface  are  a  quad- 
rant's distance   apart  on   the  great  circle   arc  through 
them,  each  is  the  pole  of  a  great  circle  through  the  other. 

Using  either  as  a  pole,  draw  the  axis,  then  'draw  the 
great  circle  of  which  it  is  the  pole. 

*  220.    If   a  point  on   the  surface  of  a  sphere  is  at  a 
quadrant's  distance  from  each  of  two  other  points  of  the 
surface,  it  is  the  pole  of  a  great  circle  through  them. 

Proceed  as  in  §  218  or  §  219.     What  cases  are  there? 

221.  Angles  between  Arcs.     The  angle  between  two  arcs 
is  the   angle  between   the  tangent   lines  at  the  point  of 
intersection. 

218.  Two  coplanar  circles  are  perpendicular  to  each  other  when 
a  radius  of  one  is  tangent  to  the  other.     Must  a  radius  of  each  arc  be 
tangent  to  the  other  circle? 

222.  Spherical   Angles.     The  most  important  class   of 


SECANTS  AND   TANGENTS 


321 


angles  between  arcs  is  that  having  great  circle  arcs  as  the 
arms  of  the  angle.  Such  an  angle  is  called  a  spherical 
angle.  As  two  great  circles  intersect  in  a  diameter,  the 
tangents  are  both  perpendicular  to  this  diameter,  and 
therefore  are  the  arms  of  a  measuring  angle  of  the  dihe- 
dral angle  between  the  great  circles. 

*  223.    The  angle  between  two  great  circle  arcs  equals 
the  measuring  angle  of  the  dihedral  angle  between  their 
planes. 

*  224.    A  great  circle  arc  through  a  pole  of  another  great 
circle  is  perpendicular  to  its  circumference. 

225.  Measurement  of  Spherical  Angles.  The  most  con- 
venient way  to  measure  a  spherical  angle  is  by  the  meas- 
uring angle  of  the  corresponding  dihedral  angle,  drawn 


A 

at  the  center  of  the  sphere.  This  angle  is  evidently  in 
the  great  circle  perpendicular  to  the  edge  of  the  dihedral 
angle,  and  therefore  perpendicular  to  the  planes  in  which 
the  arms  of  the  spherical  angle  lie. 

The  spherical  angle  XTY  equals  Z  ATB  or  Z  XOF, 
measuring  angles  of  the  dihedral  angle  between  the 
planes  TXTf  and  TYT' .  But  ZXOF  might  be  measured 
by  arc  XY.\  therefore  it  follows  that 


322          POLYHEDRAL    ANGLES   AND   THE    SPHERE 

*  226.    ji  spherical  angle  is  measured  by  the  subtended 
great  circle  arc  having  its  vertex  as  a  pole. 

219.  Explain  how  a  figure  can  be  drawn  so  that  it  will  be  bounded 
by  three  great  circle  arcs,  each  perpendicular  to  the  others. 

THE  MATERIAL  SPHERE 

*  227.    To  find  the  radius  of  a  given  circumference  on 
a  given  material  sphere. 

Select  three  points  on  the  circumference,  and  consider 
a  triangle,  with  these  three  points  as  vertices,  as  inscribed 
in  the  circle.  Construct,  in  some  plane,  the  triangle 
having  these  three  sides  (using  an  ordinary  compass  to 
carry  the  lengths,  for  it  will  measure  a  chord  of  a  circle 
on  a  sphere  just  as  well  as  a  sect  on  a  plane).  Circum- 
scribe a  circle  about  this  triangle,  and  it  will  equal  the 
given  circle.  Why  ? 

*  228.    To  describe  a  circumference  on  a  sphere  with  a 
given  pole,  and  a  given  polar  chord. 

*  229.    The  polar  chord  of  a  circle  of  a  sphere  is  the 
mean  between  the  diameter  cf  the  sphere  and  its  own 
projection  upon  the  axis  of  the  circle;  the  radius  of  a 
circle  is  the  mean  between  the  two  sects  it  makes  on  the 
axis  of  that  circle. 

Pass  a  great  circle  through  the  pole,  and  work  in  that 
plane. 

230.  Theorem  II.  To  find  the  diameter  of  a  given 
material  sphere. 

With  an}7  pole,  and  any  polar  chord  that  is  of  con- 
venient length,  describe  a  circumference  on  the  sphere ; 
find  the  radius  of  this  circle,  and  from  these  two  lengths, 
construct  the  right  triangle  considered  in  §  229. 


THE   MATERIAL  SPHERE  323 

220.  On  a  sphere  of  known  diameter,  to  construct  a  circumference 
such  that  a  triangle  inscribed  in  it  shall  be  congruent  to  a  given 
triangle. 

221.  If  the  diameter  of  a  material  sphere  is  known,  to  construct 
on  it  a  circumference  of  given  radius. 

222.  To  construct  a  circumference  on  a  given  material  sphere  so 
that  it  will  be  a  given  distance  from  the  center. 

223.  To  construct  parallel  circumferences  on  a  material  sphere. 
224-.    Given  any  circumference  on  a  sphere,  to  find  any  number  of 

points  on  the  circumference  of  the  great  circle  perpendicular  to  the 
given  circle  at  any  given  point. 

231.  Relative  Position  of  Two  Spheres.      Two   spheres 
either  lie  entirely  outside  each  other ;    lie  outside  each 
other  except  for  one  point  in  common ;    have  more  than 
one  point  in  common,  without  either  being  entirely  con- 
tained in  the  other;  or  one  is  entirely  contained  in  the 
other.     Only  the  second  and  third  of  these  possibilities 
are  of  special  interest  in  solid  geometry. 

232.  Center  Line.     The  line  through  the  centers  of  two 
spheres  is  called  their  line  of  centers.     The  sect  between 
the  centers  is  called  their  center  sect. 


233.  Tangent  Spheres.  Two  spheres  whose  surfaces 
have  one  point  in  common  are  called  tangent  spheres,  the 
point  of  tangency  being  called  the  point  of  contact.  They 
are  said  to  be  externally  or  internally  tangent,  according 
as  they  are  outside  each  other,  or  one  is  contained  in  the 
other.  Unless  otherwise  stated,  tangent  will  be  used  to 
mean  externally  tangent. 

*234.    Two  spheres  whose  surfaces  meet  in  the  center 
line  are  tangent. 

*235.    Two  spheres  tangent  to  the  same  plane  at  the 
tame  point  are  tangent. 


324          POLYHEDRAL   ANGLES   AND   THE   SPHERE 

236.  Theorem  III.  If  the  surfaces  vf  two  spheres 
meet  at  a  point  not  on  the  center  line,  they  meet  in  the 
circumference  of  a  circle  that  is  perpendicular  to  the 
center  line. 

If  a  plane  is  passed  through  the  point  and  the  two  cen- 
ters, the  two  great  circles  in  which  this  plane  intersects 
the  spheres  intersect  in  two  points  and  have  a  common 
chord.  Rotate  these  circles  to  generate  the  spheres,  and 
examine  the  figure  generated  by  their  common  points. 

225.  If  one  sphere  is  inside  another,  and  their  surfaces  meet  on  the 
center  line,  the  spheres  are  tangent. 

226.  If  the  center  sect  of  two  spheres  is  less  than  the  sum,  but 
greater  than  the  difference  of  the  radii,  the  surfaces  of  the  spheres 
intersect  in  the  circumference  of  a  circle. 

227.  If  the  center  sect  of  two  spheres  is  less  than  the  difference  of 
the  radii,  how  must  the  spheres  lie? 

228.  If  two  equal  spheres  meet,  the  circle  in  whose  circumference 
their  surfaces  intersect  is  equidistant  from  their  centers. 

229.  Find  the  center  of  a  sphere  whose  surface  passes  through  the 
circumference  of  a  given  circle,  and  contains  a  given  point  not  co- 
planar  with  the  circle. 

230.  If  from  a  point  within  a  sphere,  three  chords  are  drawn,  each 
perpendicular  to  the  other  two,  the  sum  of  the  squares  of  the  sects  of 
the  chords  equals  half  the  sum  of  the  squares  of  the  diameters  of  the 
circles  determined  by  the  chords. 


SECTION   II.     SPHERICAL   POLYGONS   AND 
POLYHEDRAL   ANGLES 

237.  Spherical  Polygons.     A  portion  of  a  spherical  sur- 
face bounded  by  three  or  more  arcs  of  great  circles  is 
called  a  spherical  polygon.     The  bounding  arcs  are  the 
sides  of  the  polygon,   the  spherical  angles  between  the 
sides  are  its  angles,  and  the  vertices  of  the  angles  are  its 
vertices. 

238.  Convex  Polygons.     A  spherical  polygon  is  convex 
if  no  side  when  extended  can  cut  the  surface  of  the  poly- 
gon.    This  is  sometimes  stated,  a  spherical  polygon   is 
convex  if  when  any  side  is  extended  to  form  a  circum- 
ference, the  whole  polygon  lies  on  the  surface  of  one  of 
the  hemispheres  bounded  by  this  circle.     Unless  other- 
wise  stated,  only  convex   polygons  will   be   considered, 
and  the  word  "convex"  should  be  understood  whenever 
necessary. 

239.  Spherical  Triangles.     A  spherical  polygon  of  three 
sides  is  called  a  spherical  triangle.     The  meanings  of  isos- 
celes, equilateral,  scalene,  right-angled,  and  any  other  such 
terms,  are  the  same  as  when  applied  to  plane  triangles. 

240.  Relation  between  Polyhedral  Angles  and  Spherical 
Polygons.     (See  §§  99-102.) 

Any  spherical  polygon  is  subtended  by  a  polyhedral 
angle  whose  vertex  is  at  the  center  of  the  sphere,  for  its 
sides  are  arcs  of  great  circles  whose  planes  intersect  at 
the  center,  and  so  bound  a  polyhedral  angle.  Conversely, 

325 


326 


POLYHEDRAL   ANGLES   AND   THE   SPHERE 


any  polyhedral  angle  whose  vertex  is 
at  the  center  of  a  sphere  subtends  a 
spherical  polygon  on  the  surface,  for 
its  faces  cut  the  surface  in  great  circle 
arcs,  which  are  the  sides,  and  its 
edges  cut  the  surface  in  points,  which 
are  the  vertices,  of  such  a  polygon. 
0-ABCD,  and  o-AfB'cfD'  are  poly- 
hedral angles  ^subtending  the  spheri- 
cal polygons  ABCD  and  A'B'C'D' . 

241.  Central  Polyhedral  Angles.  A  polyhedral  angle 
whose  vertex  is  at  the  center  of  a  sphere  is  called  a  cen- 
tral polyhedral  angle,  and  the  subtended  spherical  polygon 
is  spoken  of  as  its  polygon. 

*  242.    Each  face  angle  of  a  central  polyhedral  angle  is 
measured  by  the  subtended  side  of  its  spherical  polygon. 

*  243.    Each   dihedral  angle   of  a    central    polyhedral 
angle  has  the  same  measure  as  the  corresponding  spher- 
ical angle  of  its  spherical  polygon. 

244.  Corresponding  Propositions  on  Polyhedral  Angles 
and  Spherical  Polygons.  From  the  relations  existing  be- 
tween the  parts  of  a  central  polyhedral  angle  and  the 
parts  of  its  spherical  polygon,  it  is  evident  that  for  every 
proposition  concerning  the  face  angles  and  the  dihedral 
angles  of  a  polyhedral  angle,  there  must  be  a  correspond- 
ing proposition  concerning  the  sides  and  the  angles  of  a 
spherical  polygon,  and  conversely. 

In  the  propositions  dealing  with  these  two  kinds  of 
figures,  this  relation  is  shown  by  stating  the  propositions 
in  pairs,  one  about  polyhedral  angles,  the  other  about 
spherical  polygons.  The  one  that  is  more  convenient  to 


SPHERICAL   POLYGONS 


327 


prove  independently  is  stated  first,  the  other  follows  from 
the  relations  shown  in  §§  242  and  243. 

245.  Theorem  IV.  (V)  The  sum  of  any  two  face  angles 
of  a  trihedral  angle  is  greater  than  the  third  face 
angle. 

(ft)  The  sum  of  any  two  sides  of  a  spherical  triangle 
is  greater  than  the  third  side. 


FIKST  PROOF:  Let  the  trihedral  angle  be  V-ABC. 
Pass  a  plane  through  CV  perpendicular  to  plane  AVB, 
meeting  it  in  c'v.  Then  c'v  contains  the  projections  of 
BV  and  AV  upon  the  plane  CVC1 '.  Why  ? 

Therefore  Z  AVC1  <  Z.  AVC,  Z  BVC'  <  Z  BVC.  Why  ? 
And  Z  AVC'  +  Z  BVC'  <  Z  AVC  +  Z  BVC.  But  Z  AVC'  + 
Z  BVC'  =,  or  >,  Z  ^4F£,  according  as  VCf  falls  within  or 
outside  the  angle  AVB,  so  Z.  AVB  <  Z.  AVC  +  Z.  BVC. 

SECOND  PROOF:  Let  the  trihedral  angle  be  V-KLM, 
and  let  Z  KVL  be  its  largest  face  angle.  On  this  angle, 
cut  off  Z  RVL  =  Z  LMV.  Cut  off  VX  on  VR,  and  VC =  VX 
on  VM.  Pass  a  plane  through  X  and  C  cutting  VK  at  A 
and  FI*  at  JB.  Since  Z  #FC  =  Z  XF.B,  it  is  only  necessary 
to  prove  that  Z.  CVA  >  Z  AVX  (by  comparing  their  tri- 
angles) to  obtain  the  required  conclusion. 

Explain  in  full  why  (ft)  follows  from  (#). 

231.  Any  side  of  a  spherical  triangle  is  greater  than  the  difference 
of  the  other  two  sides.  To  what  proposition  on  polyhedral  angles 
does  this  correspond  ? 


328          POLYHEDRAL   ANGLES   AND   THE  SPHERE 

246.  COR.    If  two   circles    on    a   sphere   meet    at    a 
poiwt  on  the  great  circle  arc  through  their  poles,  they 
cannot  meet  again. 

For,  if  they  did,  on  drawing  the  polar  distances  to  that 
point,  the  sum  of  two  sides  of  the  triangle  formed  would 
equal  the  third  side.  Could  two  circles  meet  at  a  second 
point  on  the  circumference  of  the  same  great  circle  ? 

247.  Theorem  V.    The  shortest  line  from  one  point  to 
another  on  the  surface  of  a  sphere  is  the  minor  arc  of 
the  great  circle  through  them. 


Let  PQ  be  the  minor  arc  of  a  great  circle  through  the 
given  points  P  and  Q ;  then  the  shortest  line  from  P  to  Q 
on  the  surface  must  contain  every  point  of  PQ  (there- 
fore must  be  PQ),  for  if  there  is  even  one  of  the  points  of 
PQ  that  it  does  not  contain,  it  can  be  made  still  shorter. 

Suppose  the  shortest  line  from  P  to  Q  does  not  contain  T. 
With  P  and  Q  as  poles,  and  PT  and  QT  as  polar  distances, 
describe  circles,  which  will  therefore  meet  at  T.  Then 
the  circles  meet  only  at  T,  and  any  line  from  P  to  Q  not 
through  T  must  intersect  the  circles  at  different  points  E 
and  L.  But  if  circles  P  and  Q  were  rotated  so  that  JT  and  L 
were  at  T,  the  line  PKLQ  would  be  shortened  by  the  length 
KL,  its  other  parts  remaining  the  same.  Therefore  any 
line  not  containing  T  is  not  the  shortest  line  from  p  to  Q. 


SPHERICAL   POLYGONS  329 

248.  Distance  on  a  Spherical  Surface.     The  distance  be- 
tween two  points  on  a  spherical  surface  means  the  length 
of  the  minor  arc  .of  the  great  circle  through  them,  because 
the  great  circle  arc  is  the  shortest  line  between  two  points 
on  the  surface  of  a  sphere  and  therefore  corresponds  to 
the  straight  line  on  a  plane  surface.     Unless  otherwise 
stated,  only  great  circle  arcs  will  be  used. 

249.  Theorem  VI.    (a)  The  sum  of  the  face  angles  of 
any  polyhedral  angle  is  less  than  a  perigon. 

(b)    The  sum  of  the  sides  of  any  spherical  polygon  is 
less  than  the  circumference  of  a  great  circle. 

VA 


Let  ABCDEF  be  a  plane  section  of  the  polyhedral  angle 
with  vertex  F.  Then  atA,^.FAV  +  Z  VAE  >  Z.  FAB  of  the 
the  polygon  ABCDEF  (why  ?),  and  similarly  at  each  ver- 
tex of  the  polygon.  Therefore  the  sum  of  the  angles  in  the 
faces  of  the  polyhedral  angle  at  A,  B,  C,  •  •  •>  (n  —  2)  st. 
angles.  Why  ?  But  the  sum  of  all  the  angles  in  the  face 
triangles  =  ?  Therefore  the  sum  of  the  angles  at  V  =  ? 

Or,  join  a  point  O  within  ABCDEF  to  its  vertices.  The 
faces  of  the  polyhedral  angle  contain  the  same  number  of 
triangles  as  ABCDEF,  and  so  the  sum  of  the  angles  is  the 
same.  But  the  angles  in  the  faces  at  A,  J5,  C,  D,  E,  F  are 
greater  than  the  angles  in  the  polygon.  What  follows  ? 


330         POLYHEDRAL   ANGLES   AND   THE   SPHERE 

Explain  fully  why  (b)  follows  from  (a).  If  it  is  pre- 
ferred, (5)  can  be  proved  independently,  and  (a)  can  be 
deduced  from  it,  the  proof  for  any  case  following  the 
method  shown  here  for  a 
quadrilateral. 

To  show  that  the  sum 
of  the  sides  of  any  quadri- 
lateral ABCD  is  less  than  a 
circumference,  extend  AB 
and  AD  to  their  other  intersection  A  .  Extend  DC  to  meet 
AA'  at  x.  Now  use  BC<BX+XC\  XC+CD<XA'  +A' D, 
to  show  that  the  perimeter  of  ABCD  is  less  than  AAf  +  AfA, 
or  a  circumference. 

232.  If,  in  the  figure  of  Theorem  VI,   V  approaches  indefinitely 
near  to  the  plane  ABCDEF,  what  change  is  there  in  the  sum  of  the 
angles  at  F?  what  change  if   V  moves  indefinitely  far  away?     Be- 
tween what  limits  might  it  be  said  that  the  sum  of  the  face  angles  of 
a  polyhedral  angle  must  lie  ? 

233.  Between  what  limits  might  the  sum  of  the  sides  of  a  spheri- 
cal polygon  be  said  to  lie? 

234-  Use  the  method  shown  in  (6)  of  Theorem  VI  to  prove  the 
sum  of  the  sides  of  a  triangle  less  than  a  circumference ;  of  a  hexagon. 

235.  Prove,  without  using  Theorem  V,  that  the  great  circle  arc 
between  two  points  is  less  than  any  other  line  between  those  points 
made  up  of  sects  of  great  circle  circumferences. 

236.  If  from  the  ends  of  a  side  of  a  spherical  triangle,  great  circle 
arcs  are  drawn  to  a  point  within  the  triangle,  the  sum  of  those  arcs 
is  less  than  the  sum  of  the  other  two  sides  of  the  triangle. 

250.  Regular  Polyhedrons.  A  polyhedron  is  said  to  be 
regular  if  its  faces  are  all  regular  polygons  of  the  same 
number  of  sides.  Since  each  edge  is  common  to  two  faces, 
this  means  that  all  the  edges  are  equal,  and  that  all  the 
faces  are  congruent.  It  will  be  evident  from  the  descrip- 


SPHERICAL   POLYGONS  331 

tions  of  the  regular  polyhedrons  that  all  the  polyhedral 
angles  of  a  regular  polyhedron  are  congruent,  and  that 
all  its  dihedral  angles  are  therefore  equal. 

251.  Theorem   VII.     There  cannot  be  more  than  five 
regular  polyhedrons. 

The  sum  of  the  face  angles  of  a  polyhedral  angle  is  less 
than  360°.  Why?  Since  an  angle  of  an  equilateral  tri- 
angle is  60°, there  can  be  three,  four,  or  five  such  face  angles. 
Six  such  angles  around  a  point  would  lie  in  a  plane,  and 
seven  or  more  would  be  impossible.  The  angle  of  a  square 
is  90°,  so  there  could.be  three  such  face  angles.  Four 
would  lie  in  a  plane,  and  five  or  more  would  be  impossible. 
An  angle  of  a  regular  pentagon  is  108°,  so  there  could  be 
three  such  face  angles,  but  no  more.  Three  hexagons 
would  lie  in  a  plane,  four  would  be  impossible ;  and  three 
polygons  of  any  number  of  sides  greater  than  six  would 
be  impossible.  Therefore  solid  angles  .can  be  formed 
by  three,  four,  or  five  equilateral  triangles,  three  squares, 
or  three  regular  pentagons,  but  in  no  other  way  from  reg- 
ular polygons. 

It  is  possible  (by  following  construction  methods),  to 
prove  that  a  regular  polyhedron  can  be  constructed  by 
using  each  of  these  possibilities.  The  method  will  be 
shown,  and  the  polyhedrons  will  be  described  and  named. 

252.  The  Regular   Tetrahedron.     The   regular  tetrahe- 
dron is  a  solid  bounded  by  four  equilateral  triangles.     It 
could  be  constructed  by  erecting  a  perpendicular  at  the 
circumcenter  of  an  equilateral  triangle,  and  cutting  this 
line  by  a  circle  around  one  of  the  vertices  as  a  center, 
with  the  side  as  a  radius.     The  point  so  found  would  be 
the  fourth  vertex  of  the  tetrahedron,  which  is  evidently 


332 


POLYHEDRAL   ANGLES   AND   THE   SPHERE 


a  triangular  pyramid.     It  is  the  polyhedron  having  three 
equilateral  triangles  about  each  vertex. 

As  the  faces  of  the  regular  tetrahedron 
are  equilateral  triangles,  it  is  not  diffi- 
cult to  find  its  area  and  volume  in 
terms  of  the  side.  Given  the  edge  E, 
the  total  area  is  .E2V3.  To  find  the 
volume,  the  altitude  is  necessary,  and  it 
can  be  found  by  use  of  a  right  triangle  ;  as,  in  the  figure, 

VC  =  E,  OC 


-  the  median  or  altitude,  =  —  x  -  V3=  -  V3; 
o  3      2  o 


therefore  VO  =  —  V6,  and  the  volume  =  —  V6. 

253.    The  Regular  Hexahedron  or  Cube. 

The  polyhedron  having  three  squares 
around  each  vertex  is  a  regular  hexa- 
hedron, or  a  cube.  It  is  a  regular  rec- 
tangular parallelepiped. 

If  the  edge  is  E,  its  surface  is  6  E2, 
and  the  volume  is  EB;  the  diagonal  of 
a  face  is  ^V2,  and  of  the  cube  is  ^V3.  Why  ? 

237.  If  the  diagonal  of  a  cube  is  8  in.,  find  its  volume. 

238.  What  is  the  length  of  an  edge  of  a  cube,  if  the  cube  is  twice 
as  large  as  the  cube  of  edge  3  in.  ? 

239.  Given  two  cubes,  to  construct  a  cube  whose  area  is  the  sum 
of  their  areas. 

NOTE.  It  is  not  possible,  by  using  the  straight  edge  and  the  com- 
pass, to  add  the  volumes  of  cubes,  or  to  multiply  the  volume  of  a 
cube,  obtaining  another  cube  as  the  result.  This  means  that  the 
same  rule  is  true  for  the  other  solids,  so  while  it  is  possible  to  con- 
struct a  sphere  or  a  regular  tetrahedron  such  that  its  area  shall  be 
the  sum  of  the  areas  of  two  figures  of  the  same  kind,  it  is  not  possible 
to  perform  the  same  operation  on  their  volumes. 

The  problem  of  duplication  of  the  cube,  that  is,  of  the  construction 
(with  straight  edge  and  compasses)  of  a  cube  of  volume  double  that 


SPHERICAL  POLYGONS  333 

of  a  given  cube,  ranks  with  the  squaring  of  the  circle  and  the  trisec- 
tion  of  an  angle  as  one  of  the  geometrical  impossibilities,  on  which 
mathematicians  worked  for  centuries.  Any  history  of  mathematics 
will  give  an  interesting  account  of  these  attempts. 

240.    To  construct  a  cube  having  a  given  edge. 

254.  The  Regular  Octahedron.  The  solid  having  four 
equilateral  triangles  around  each  vertex  is  a  regular 
octahedron.  It  might 
be  described  as  two 
square  pyramids  whose 
edges  are  all  equal, 
placed  base  to  base. 

If    the     octahedron 
has  the  edge  E,  its  sur- 
face   is    2j;2V3.     To 
obtain  the  volume,  drop  a  perpendicular  from  X  to  ABCD 
at  O.    Then  AXC  is  an  isosceles  triangle  with  sides  CX=E, 

XA  =  E,  AC  =  #V2  ;    therefore   Z  CXA  is   a   right   angle 

X 
(why?),  and  XO  and  AO  are  each  ~oV2.     The  volume  is 


241.  To  construct  the  regular  octahedron  of  edge  E. 

242.  Find  the  volume  of  a  regular  octahedron  of  edge  5  in. 

243.  If  the  diagonal  of  a  regular  octahedron  is  12  in.,  find  its  area. 

244.  Use  the  prismatoid  formula    to    find  the    volume   of   the 
octahedron. 

245.  Show  that  the  lines  joining  the  midpoints  of  the  adjoining 
faces  of  a  cube  are  the  edges  of  a  regular  octahedron,  and  find  the 
ratio  of  its  area  and  volume  to  the  area  and  volume  of  the  cube. 

255.  The  Regular  Dodecahedron.  The  solid  having 
three  regular  pentagons  around  each  vertex  is  a  regular 
dodecahedron.  It  might  be  described  as  a  solid  having 


334          POLYHEDRAL   ANGLES  AND   THE   SPHERE 

two  regular  pentagons  as  parallel  bases,  the  other  faces 
being  ten  regular  pentagons,  arranged  five  around  each 
of  the  bases. 


256.  The  Regular  Icosahedron.  The  solid  having  five 
equilateral  triangles  about  each  vertex  is  a  regular  icosahe- 
dron. It  might  be  described  as  two  pyramids  with  penta- 


gons as  bases,  connected  by  ten  equilateral  triangles,  each 
having  for  its  base  a  side  of  one  pentagon,  and  for  its 
vertex,  a  vertex  of  the  other  pentagon. 

Neither  the  dodecagon  nor  the  icosahedron  is  of  much 
importance  in  elementary  geometry,  but  they,  as  well  as 
the  other  regular  polyhedrons,  are  of  importance  in  crys- 
tallography. 

246.  How  many  edges  and  vertices  have  the  regular  dodecagon 
and  icosahedron? 


SPHERICAL   POLYGONS  335 

257.  Vertical   Polyhedral   Angles.      If   the  planes  that 
bound  a  polyhedral  angle  are  extended  through  the  vertex, 
they  bound  a  second  polyhedral  angle  that  is  called  vertical, 
or  opposite,  to   the   first  angle.     In  the  figure  of   §  240, 
O-ABCD  and  o-ArBfCfDr  are  vertical  polyhedral  angles. 

258.  Opposite  Spherical  Polygons.     Spherical  polygons 
subtended  by  vertical  central  polyhedral  angles  are  called 
opposite  spherical  polygons.     In  the  figure  of  §  240,  ABCD 
and   A'B'C'D'    are    opposite   spherical   polygons.     If   the 
sides  of  opposite   spherical   polygons  that  are  subtended 
by  vertical  face  angles  at  the  center  are   considered  as 
corresponding  sides,  and  the  spherical  angles  having  the 
ends  of  a  diameter   as  their   vertices  are   considered  as 
corresponding  angles,  it  is  clear  that  corresponding  sides 
are  arcs  of  the  same  great  circle  circumference,  and  that 
corresponding  spherical  angles  are  formed  by  the  same 
two  planes. 

259.  Division  of  the  Surface  by  Circumferences  of  Great 
Circles.     The  circumferences  of  two  great  circles  meet  at 
the  ends  of  a  diameter,  and  divide  the  surface  into  four  parts 
called  lunes.       (See  also  §285.)     A  third  great  circum- 
ference, not  through  either  end  of  the  same  diameter,  will 
cut  each  of  these  circumferences  twice,  thus  dividing  each 
of  the   lunes  into  two  triangles,  and  the  surface   of  the 
sphere  into  eight  triangles.     These  triangles  are  evidently 
four  pairs  of  opposite  triangles,  since  they  are  formed  by 
the  same  great  circle  planes. 

It  follows  that,  if  the  sides  of  any  spherical  triangle  are 
extended,  they  divide  the  surface  of  the  sphere  into  eight 
triangles,  one  of  which  is  opposite  to  the  given  triangle. 

260.  Symmetric    Spherical    Polygons    and    Polyhedral 
Angles.     Two    spherical    polygons    are    called    symmetric 


336         POLYHEDRAL   ANGLES  AND   THE   SPHERE 

when  their  corresponding  sides  are  equal,  and  their 
corresponding  angles  are  equal,  the  parts  being  arranged 
in  opposite  order,  in  the  two  polygons.  For  example,  if  a 
polygon  has  the  sides  a,  5,  c,  d,  arranged  in  the  order  in- 
dicated when  read  in  the  direction  taken  by  the  hands  of 
a  clock  {clockwise),  its  symmetric  polygon  will  have  its 
sides  in  this  order  when  read  in  the  opposite  direction, 
that  is,  counter-clockwise. 

Similarly,  polyhedral  angles  are  symmetric  when  their 
corresponding  parts  are  equal,  but  are  arranged  in  oppo- 
site order. 

*  261.    Vertical  polyhedral  angles  are  symmetric.     See 
the  figure  of  §  240.     Show  that  the  parts  are  correspond- 
ingly equal,  then  examine  the  order,  viewed  from  the 
common  vertex. 

*  262.    Opposite  spherical  polygons  are  symmetric.    See 
the  figure  of  §  240.    View  the  polygons  from  the  center  of 
the  sphere  ;  otherwise  the  order  may  become  confused. 

263.  Coincidence  of  Great  Circle  Arcs  and  of  Spherical 
Angles.  The  circumferences  of  great  circles  coincide  if 
they  have  two  points  other  than  the  ends  of  a  diameter 
in  common,  and  if  they  have  the  same  center.  In  other 
words,  two  points  on  the  surface  of  a  sphere  determine 
the  circumference  of  a  great  circle.  (Why  ?)  Therefore, 
if  one  of  two  great  circle  arcs  is  placed  along  another,  with 
one  end  in  common,  their  centers  being  the  same,  it  will 
lie  along  the  other  throughout  its  length.  Therefore  the 
first  arc  will  coincide  with  the  second  if  the  two  arcs  are 
equal ;  the  first  will  include  the  second  if  the  first  is  the 
longer ;  and  the  first  will  be  included  by  the  second  if  the 
first  is  the  shorter. 


SPHERICAL   POLYGONS  337 

If  one  of  two  spherical  angles  is  placed  on  another  (the 
center  of  their  spheres  being  the  same),  with  their  vertices 
coinciding,  and  one  arm  of  the  first  angle  along  the  corre- 
sponding arm  of  the  other  angle,  then  the  other  arm  of  the 
first  angle  must  also  fall  along  its  corresponding  arm  if 
the  angles  are  equal  (why  ?),  so  making  the  angles  coin- 
cide ;  it  must  fall  outside  the  corresponding  arm  if  that 
angle  is  the  greater ;  and  it  must  fall  inside  the  correspond- 
ing arm  if  that  angle  is  the  smaller. 

In  placing  either  an  arc,  or  a  spherical  angle,  on  another, 
it  must  be  considered  as  being  slid  along  the  surface  into 
the  required  position,  for  if  it  is  turned  over,  that  is,  if 
the  position  of  the  center  of  the  sphere  is  changed,  the 
curvature  is  reversed,  and  the  arcs  are  no  longer  on  the 
same  sphere,  and  cannot  coincide.  This  is  most  easily 
shown  by  the  arcs 
of  circles  in  a  plane, 
the  same  reasoning 
applying  for  the 
arcs  on  a  sphere. 
Let  the  circles  O 
and  Or  be  equal, 
and  let  arc  A B  equal 
arc  XY.  Although 
they  have  two  points  in  common,  they  do  not  coincide,  for 
they  are  not  on  the  circumference  of  the  same  circle.  Now 
suppose  circle  Of  to  be  superposed  on  circle  O  with  the 
centers  coinciding,  and  so  that  arc  XY  lies  along  arc  AB 
with  one  end  in  common.  Then  in  this  position  they 
coincide.  Similarly,  if  arc  KL  equals  arc  AB,  it  coincides 
with  it  if  it  is  supposed  to  slide  along  the  circumference 
until  ,L  falls  on  A.  It  cannot  coincide  with  AB  if  it  is 
turned  over  so  that  its  center  is  no  longer  at  O. 


338         POLYHEDRAL   ANGLES   AND   THE   SPHERE 

264.  Congruence  and  Symmetry  of  Spherical  Polygons. 
Two  spherical  polygons  are  congruent  if  they  can  be  made 
to  coincide.  If  they  can  coincide,  their  corresponding  (or 
homologous)  parts  must  be  equal,  and  arranged  in  the  same 
order.  Conversely,  if  their  corresponding  parts  are  equal, 
and  arranged  in  the  same  order,  they  can  be  superposed, 
and  by  examining  the  position  of  each  vertex  and  side, 
they  can  be  shown  to  coincide.  If,  however,  the  parts  are 
equal,  but  arranged  in  opposite  order,  the  figures  cannot 
be  made  to  coincide,  that  is,  symmetric  polygons  are  not 
congruent.  (See  §  272.) 

If  in  A  ABC,  A  DEF,  and  A  GHK,  AB  =  DE  =  GH,  EC  = 
EF=KG,  CA  =  FD  =  HK,  and  correspondingly  for  the 
angles,  the  triangles  are  not  all  con- 
gruent, for  the  parts  of  A  GHK  are 
arranged  in  opposite  order  to  that 
in  which  the  parts  of  A  ABC  and 
A  DEF  are  arranged,  and  so  A  GHK 
cannot  be  superposed  on  one  of  them 
by  sliding  it  along  the  surface, — 
as  could  be  done  with  A  ABC  and 
A  DEF,  —  but  must  be  turned  over, 
that  is,  placed  with  the  center  of  its  sphere  outside  of 
sphere  O  instead  of  at  center  O,  to  bring  its  parts  in  the 
right  order.  But  this  reverses  the  curvature  of  its  sides, 
—  that  is,  they  are  no  longer  great  circle  arcs  on  the  same 
sphere,  —  so  they  cannot  be  made  to  coincide,  and  the  tri- 
angle will  not  coincide  with  either  of  the  other  triangles, 
and  so  is  not  congruent  to  them. 

Note  that  polygons  symmetric  to  the  same  polygon  are 
congruent. 

Explain  why  two  plane  triangles  are  congruent  if  their 
parts  are  equal  but  arranged  in  opposite  order. 


SPHERICAL   POLYGONS  339 

265.  Congruence  and  Symmetry  of  Polyhedral  Angles. 
Polyhedral  angles,  like  spherical  polygons,  are  congruent 
when  they  can  be  made  to  coincide.     They  can  be  made 
to  coincide  when  their  corresponding  parts  are  equal  and 
arranged  in  the  same  order  ;   but  they  are  only  symmetric 
when  the  parts  are  equal,  but  are  arranged  in  opposite 
order. 

Examples  of  symmetric  bodies,  that  is,  of  bodies  that 
are  equal  in  all  respects,  but  cannot  coincide  because 
rtheir  parts  are  arranged  in  opposite  order,  are  numerous. 
Among  the  most  familiar  examples  are  one's  hands,  feet, 
ears,  shoulders,  a  right  and  a  left  glove,  a  pair  of  shoes, 
the  inside  and  the  outside  of  a  door,  the  two  halves  of  a 
piano  if  cut  from  front  to  back,  the  two  blades  of  a  pair 
of  scissors,  etc. 

266.  Theorem  VIII.     (#)  Two  triangles  on  the  same 
sphere  are  either  congruent  or  symmetric  if  they  have 
two   sides  and   the  included   angle  of  one  respectively 
equal  to  two  sides  and  the  included  angle  of  the  other. 

(5)  Two  trihedral  angles  are  either  congruent  or  sym- 
metric if  they  have  two  face  angles  and  the  included 
dihedral  angle  of  one  respectively  equal  to  two  face  angles 
and  the  included  dihedral  angle  of  the  other. 

Superpose  if  the  parts  are  arranged  in  like  order.  If 
not,  superpose  one  on  the  triangle  that  is  opposite  to  the 
other. 

267.  Theorem  IX.     (a)  Two  triangles   on  the  same 
sphere  are  either  congruent  or  symmetric  if  they  have 
two   angles   and  the   included  side  of  one  respectively 
equal  to  two  angles  and  the  included  side  of  the  other. 

(b)  Two  trihedral  angles  are  either  congruent  or  sym- 
metric if  they  have  two  dihedral  angles  and  the  in- 


340          POLYHEDRAL   ANGLES   AND   THE   SPHERE 

eluded  face  angle  of  one  equal  to  two  dihedral  angles 
and  the  included  face  angle  of  the  other. 

247.  Prove  Theorem  IX  without  superposing. 

268.  Theorem  X.      (a)  In  an  isosceles  spherical  tri- 
angle, the  base  angles  are  equal. 

(5)  If  two  face  angles  of  a  trihedral  angle  are  equal, 
the  opposite  dihedral  angles  are  equal. 

Prove  as  in  plane  geometry. 

269.  COR.  1.     In  an  isosceles  spherical  triangle,  the 
bisector  of  the  vertex  angle,  the  median  to  the  base,  the 
altitude,  and  the  perpendicular  bisector  of  the  base,  are 
all  one  line. 

270.  COR.  2.     Two  isosceles  symmetric  spherical  tri- 
angles are  congruent. 

248.  Two  spherical  right  triangles  are  congruent  or  symmetric  if 
they  have  two  sides  of  one  equal  to  the  corresponding  sides  of  the 
other. 

271.  Theorem  XI.     (a)    Two  triangles  on  the    same 
sphere  are  either  congruent   or  symmetric  if  they  have 
the  three  sides   of  one  equal  respectively  to  the   three 
sides  of  the  other. 

(£>)  Two  trihedral  angles  are  either  congruent  or 
symmetric  if  they  have  the  three  face,  angles  of  one 
respectively  equal  to  the  three  face  angles  of  the 
other. 

272.  Theorem  XII.     Two   symmetric   spherical   trian- 
gles are  equivalent. 

Draw  a  circle  through  the  vertices  of  each  triangle  and 
prove  the  circles  equal.  Draw  the  polar  distances  to  the 
vertices  of  the  triangles,  and  prove  them  equal.  Show  that 


SPHERICAL  POLYGONS  341 

the  three  triangles  thus  formed  in  one  of  the  given  tri- 
angles are  respectively  congruent  to  the  three  triangles 
formed  in  the  other  triangle, — in  other  words,  that  sym- 
metric triangles  can  be  divided  into  congruent  parts,  and 
so  are  equivalent. 

249.  The  median  to  the  base  of  an  isosceles  spherical  triangle 
divides  the  triangle  into  equivalent  triangles. 

250.  If  the  opposite  sides  of  a  spherical  quadrilateral  are  equal, 
either  diagonal  divides  it  into  equivalent  triangles ;  the  diagonals  bi- 
sect each  other. 

273.  Polar  Triangles.  If  for  each  side  of  a  spherical 
triangle  that  one  of  its  poles  is  taken  that  is  on  the  same 
hemispherical  surface  as  the  opposite  vertex  (the  circle 
of  which  that  side  is  an  arc  being  considered  to  form  the 
hemispheres),  and  if  these  three  points  are  joined  by  minor 
arcs  of  great  circles,  the  resulting  triangle  is  called  the 
polar  triangle  of  the  original  triangle.  In  the  figure,  if 


A* ',  Bf,  c',  are  the  poles  of  BC,  CA,  and  AB  respectively, 
since  A  and  A1  are  on  the  same  hemispherical  surface  with 
reference  to  BC,  B  and  Br  are  on  the  same  hemispherical 
surface  with  reference  to  CA,  and  C  and  cf  are  on  the  same 
hemispherical  surface  with  reference  to  AB,  then  the  great 
circle  arcs  A'B',  BfCr,  and  C'A'  form  the  polar  triangle  of 
triangle  ABC. 


342       POLYHEDRAL   ANGLES   AND  THE   SPHERE 

274.  Theorem  XIII.     If  the  first  of  two  spherical  tri- 
angles is  the  polar  triangle  of  the  second,  the  second  is 
also  the  polar  triangle  of  the  first. 

How  can  a  point  be  proved  the  pole  of  an  arc  ?  No 
construction  lines  need  be  drawn. 

Two  triangles,  such  that  each  is  the  polar  of  the  other, 
are  called  polar  triangles. 

275.  Theorem  XIV.     In  two  polar  triangles,  each  angle 
of  one  is  measured  by  the  supplement  of  the  side  oppo- 
site to  it  in  the  other. 


It  is  necessary  to  prove  that  Z  c  is  measured  by  180°  — 
A'B '.  Since  C  is  the  pole  of  AfBf,  by  what  arc  can  Z  c  be 
measured?  Show  that  this  arc  is  180°  —  A'B'  by  using 
the  fact  that  A'  and  Bf  are  poles  of  the  great  circles 
through  BC  and  CA  respectively.  They  therefore  have 
what  polar  distances  to  those  circumferences? 

251.  The  polar  triangle  of  a  right  triangle  has  one  side  a  quadrant. 

252.  If  the  angles  of  a  triangle  are  135°,  97°,  and  88°,  find  the  op- 
posite sides  of  the  polar  triangle. 

253.  Show  that  if  one  side  of  a  spherical  polygon  is  more  than  180°, 
the  polygon  is  not  convex. 

254.  If  the  sides  of  two  polar  triangles  meet,  in  how  many  points 
can  they  intersect  ? 

255.  Two  triangles,  such  that  one  side  of  each  is  a  quadrant  (quad- 
rantal  triangles},  are  congruent  or  symmetric  if  they  have  two  angles 
of  one  respectively  equal  to  two  angles  of  the  other. 


SPHERICAL   POLYGONS  343 

276.  Theorem   XV.     (#)  Two  triangles  on  the  same 
sphere  are  either  congruent  or  symmetric  if  they  have 
the  three  angles  of  one  respectively  equal  to  the  three 
angles  of  the  other. 

(ft)  Two  trihedral  angles  are  either  congruent  or 
symmetric  if  they  have  the  three  dihedral  angles  of 
one  respectively  equal  to  the  three  dihedral  angles  of 
the  other. 

For  the  sides  of  their  polar  triangles  are  equal  (why  ?), 
so  the  polar  triangles  are  congruent  or  symmetric.  Show 
that  the  sides  of  the  given  triangles  must  therefore  be 
respectively  equal. 

256.  If  two  angles  of  a  spherical  triangle  are  equal,  the  opposite 
sides  are  equal.  (Use  the  polar  triangles.)  State  this  for  a  trihedral 
angle. 

277.  Theorem  XVI.     The  sum  of  the  angles  of  a  spheri- 
cal  triangle  is  greater  than  one,  and  less  than  three, 
straight  angles. 

Let  the  angles  be  A,  B,  (7,  the  opposite  sides  be  a,  ft,  c, 
and  the  angles  and  sides  of  the  polar  triangle  be  A*,  B1 ',  C', 
and  a',  br,  c'  : 

then  4  +  B  +  C  =  540°-(a'+6'  +  <O.     WhJ? 
But  a'  +  bf  4-  cf  lies  between  what  limits  ? 

278.  Spherical  Excess.     §  277  proves  that  the  sum  of 
the  angles  of  a  spherical  triangle  is  greater  than  the  sum 

i  of  the  angles  of  a  plane  triangle.  From  this  it  follows  that 
the  sum  of  the  angles  of  any  spherical  polygon  of  n  sides 
is  more  than  (n  —  2 )  straight  angles.  The  amount  by 
which  the  sum  of  the  angles  of  a  spherical  polygon  ex- 
ceeds the  sum  of  the  angles  of  a  plane  polygon  of  the  same 
number  of  sides  is  called  the  spherical  excess  of  that  poly- 
gon. The  spherical  excess  of  a  triangle  is  shown  in  §  277 
to  be  between  zero  and  two  straight  angles. 


SECTION   in.     AREA   ON   A   SPHERICAL   SURFACE 

279.  Generation  of  the  Surface   of  a  Sphere.      It  has 

been  shown  (§  188)  that  the  rotation  of  a  semicircle  about 
its  diameter  generates  a  sphere.  It  follows  that  the  sur- 
face generated  by  the  semicircumference  is  a  spherical 
surface,  and  from  this  fact  the  area  of  a  spherical  surface 
can  be  found. 

280.  Zones.     That  part  of  the  surface  of  a  sphere  which 
is  included  between    two   parallel  planes  that  meet   the 
surface  is  called  a  zone.     Either  of  the  planes  may  be  a 
tangent  or  a  secant  plane,  so  the  zone  can  be  bounded  by 
two  circumferences  on  the  surface,  by  but  one  circumfer- 
ence when  one  plane  is  tangent,  or  it  can    include    the 
whole  surface.     The  perpendicular  between  the  planes  is 
the  altitude  of  the  zone. 

The  zones  on  the  surface  of  the  earth  are  familiar  ex- 
amples. 

281.  Generation  of  a  Zone.      The  rotation  of   any  arc 
of  a  semicircle  about  its  diameter  generates  a  zone,  for 
all  points  of   the   surface    so    generated    are    equidistant f 
from  the  center  of  the  circle,  and  the  surface  lies  between 
planes  perpendicular  to  the  diameter. 

282.  If,  in  the  generating  arc  of  a  zone,  a  broJcen 
line  is  inscribed  whose  vertices  divide  the  arc  into  equal 
parts,  then,  as  the  number  of  these  parts  is  increased 
indefinitely,  the  area  generated  by  the  broken  line  ap- 
proaches the   area   of  the   zone   as   its  limit. 

344 


AREA   ON   A   SPHERICAL   SURFACE  345 

This  is  practically  an  axiom  of  the  sphere,  although 
it  is  evident  from  the  plane  geometry  discussion  of 
the  regular  polygon  inscribed  in  a  circle.  It  may  be 
called  a  proposition,  but  in  any  case  it  is  assumed  here 
without  proof. 

283.  Theorem     XVII.       The  area  of  a  zone  is  equal 
to  the  product  of  its  altitude  and  the  circumference  of 
a  great  circle-    Area  =  2  irrh. 

For,  if  a  broken  line  is  inscribed,  as  explained  in  §  282, 
the  area  generated  by  any  one  sect  will  be  2  7rahf,  where 
a  is  the  apothem  to  the  broken  line,  and  hf  is  the  altitude 
of  the  frustum  generated  by  that  sect.  (See  §  171,  the 
perpendicular  in  that  formula  being  the  apothem  in  this 
figure).  On  adding  these  areas  for  all  the  sects,  and  rep- 
resenting the  sum  of  the  altitudes,  or  the  altitude  of  the 
zone,  by  A,  the  total  area  generated  by  the  broken  line  is 
2  Trail.  But  if  the  number  of  parts  is  increased  indefinitely, 
the  surface  generated  by  the  broken  line  approaches  the 
zone,  and  the  apothem  approaches  the  radius  of  the  sphere. 
Therefore  the  area  of  the  zone  is  2  irrh. 

284.  COR.      The    area   of   a  spherical  surface  is  the 
product  of   its  diameter  and  the  circumference   of   a 
great  circle.    Area  =•  4  Trr2. 

For  it  is  a  zone  of  what  altitude  ? 

257.  Zones  on  the  same  sphere  are  proportional  to  their  altitudes. 

258.  Find  the  area  of  a  sphere  of  radius  2  inches.     How  does  it 
compare  with  the  area  of  a  great  circle  ?     How  does  the  area  of  any 
zone  compare  with  the  area  of  a  great  circle  ? 

259.  Find  the  area  of  a  zone  on  a  sphere  of  radius  1  ft.  if  the  polar 
distance  of  one  of  its  circles  is  30°  and  of  the  other  is  60°. 


346 


POLYHEDRAL  ANGLES   AND   SPHERES 


285.  Lunes.      That    part     of     a 
spherical   surface  bounded  by  two 
great  semicircumferences   is  called 
a  lune.     See  also  §  259.     The  lune 
can  be  of  any  size  from  zero  to  the 
entire  surface  of  the  sphere.     The 
two   semicircumferences  form    two 
spherical  angles  at  their  intersec- 
tions, and  these  angles  are  equal.    (Why  ?)    In  the  figure, 
two  lunes  are  shown,  one  bounded  by  PXP'  and  PFP',  the 
other  by  PYP1  and  PZP';  their  angles  are  marked  A  and  B. 

286.  Addition  and  Subtraction  of  Lunes.     Two  lunes  are 
added  by  placing  them  so  that  they  lie  on  opposite  sides 
of    a   common    semicircumference,    the    entire   figure    so 
formed  being  their  sum.     They  are  subtracted  by  placing 
them  on  the  same  side  of  a  common  semicircumference, 
that  part  of  the  larger  not  contained  in  the  smaller  being 
their  difference. 

The  sum  or  the  difference  of  two  lunes  is  also  a  lune,  — 
since  it  is  still  bounded  by  semicircumferences,  —  as  is  also 
any  multiple  of  a  lune,  for  it  could  be  formed  by  succes- 
sive additions  of  equal  lunes. 

287.  Equality  of  Lunes.     If,  on  the  same  spherical  sur- 
face, one   lune  is  placed  on  another  with  one  bounding 
semicircumference  in  common,  the  other  circumferences 
either  coincide,  or  one  of  them  lies  entirely  within  the  lune 
bounded  by  the  other.     There  is,  therefore,  no  difference 
between   equivalence  and  congruence  of   lunes,  and  the 
word  equal  will  be  used  for  them. 

288.  Notation  for  Lunes.     In  dealing  with  lunes,  the 
lune  is  designated  by  the  letter  L  with  a  subscript  denot- 
ing the  spherical  angle  at  the  point  of  intersection  of  the 


AREA   ON   A   SPHERICAL   SURFACE  347 

semicircumferences ;  for  example,  LA  denotes  the  lune 
whose  spherical  angle  is  A ;  L1SQO  denotes  the  lune  whose 
spherical  angle  is  180°,  —  in  other  words,  a  hemispherical 
surface.  That  this  is  a  satisfactory  way  in  which  to  denote 
a  lune  will  be  evident  from  the  following  paragraphs. 

289.  Relations  between  Lunes  and  their  Spherical  Angles. 
From  the  foregoing  discussion,  and  the  fact  that  a  spheri- 
cal angle  has  the  same  measure  as  the  dihedral  angle 
between  the  circles  whose  arcs  are  its  arms,  the  following 
propositions  about  lunes  follow. 

*  290.   Lunes  are  equal  if  their   spherical    angles  are 
equal. 

LA  =  LB  if  A  =  B.  Superpose. 

*  291.    If  two  lunes   are   equal,  their   spherical   angles 

are  equal. 

If  LA  =  LB,  then  A  =  B. 

*  292.    The  spherical  angle  of  the  sum,  or  of  the  differ- 
ence, of  two  lunes  is  the  sum,  or  the  difference,  respec- 
tively, of  their  spherical  angles. 

LA  ±  LB  ==  L(A±B)' 

*  293.    The  spherical  angle  of  any  multiple  of  a  lune 
is  the  same  multiple  of  its  spherical  angle. 


*  294.    Lunes   are  proportional   to    their   spherical    an- 
gles. 

L±  =  A 
LB      B 

Use  the  ordinary  method  of  proof  for  the  commensurable 
case, 


348  POLYHEDRAL   ANGLES   AND   SPHERES 

295.  Theorem  XVIII.  The  area  of  a  lune  is  the  same 
part  of  the  area  of  the  sphere  that  its  spherical  angle 
is  of  360°. 


260.  Find  the  area  of  a  lune  of  angle  20°  on  a  sphere  of  radius 
18  inches. 

261.  Find  the  angle  of  a  lune  that  is  one  fifth  of  the  spherical 
surface. 

262.  Find  the  angle  of  a  lune  that  is  equivalent  to  a  trirectangular 
triangle  (one  whose  angles  are  all  right  angles). 

263.  A  lune  on  a  sphere  of   radius  10  in.  is  equivalent   to  the 
whole  surface  of  a  sphere  of  radius  4  m.     What  is  its  angle  ? 

296.  Theorem  XIX.  A  spherical  triangle  is  equiva- 
lent to  a  lune  whose  angle  is  one  half  the  spherical 
excess  of  the  triangle. 

Call  the  triangle  ABC,  T.  Let  its 
spherical  excess  be  E,  and  letter  the 
intersections  of  the  great  circum- 
ferences that  form  its  sides  as  in 
the  figure.  Then  the  semisurface 
ACrAfC  (or  £180o)  is  composed  of 
four  triangles,  whose  areas  are  as 
follows : 

A  ABC  =   T;   A  ACf B  =  Lc  —  T;   A  A' CB  =  LA  —  T; 

A  A'BC'  =  A  AB'C  (opposite  triangles), .'.  =  LB  —  T. 
Therefore  on  adding  these  values  of  the  four  triangles, 
L18QO  =  LA  +  LB  +  Lc  —  2  T,  and  on  solving  for  T, 

T  —  L180°-A-B-C  =  LE' 


AREA  'ON   A   SPHERICAL   SURFACE  349 

297.  COR.  1.  The  area  of  a  spherical  triangle  is  the 
same  part  of  the  area  of  the  sphere  that  its  spherical 
excess  is  of  720°. 


298.  COR.  2.  The  area  of  a  spherical  polygon  is  the 
same  part  of  the  area  of  the  sphere  that  its  spherical 
excess  is  of  720°. 


299.  Notation  for  Spherical  Triangles  and  Spherical 
Polygons.  In  finding  the  areas  of  spherical  polygons, 
since  only  the  angles  are  used,  a  convenient  notation  is  to 
denote  a  triangle  by  T  with  a  subscript  to  show  its  angles. 
Thus,  T(AtBtC)  would  represent  the  triangle  of  angles  A,  5, 
and  <7,  and  T(970>  850(  103o)  would  represent  the  triangle  having 
angles  of  97°,  85°,  and  103°.  It  is  convenient  also  to 
denote  a  polygon  of  more  than  three  sides  by  P  with  a 
subscript  to  show  its  angles. 

264-  On  a  sphere  of  radius  10  ft.,  find  the  area  of  (a)  a  triangle 
of  angles  84°  30',  111°  28',  35°  17'  ;  (ft)  a  pentagon  each  angle  of  which 
is  120°. 

265.  If  an  equiangular  triangle  is  one  tenth  of  the  spherical  sur- 
face, find  its  angles. 

266.  A  spherical  triangle  whose  angles  are  in  the  ratios  1:2:3  is 

equivalent  to  a  zone  of  altitude  -.     Find  its  angles. 

3 

267.  A  spherical  polygon   on  a  sphere  of  radius  6  in.  has  each 
angle  160°,  and  its  area  is  44  sq.  in.     How  many  sides  has  it? 

268.  Find  the  area  of  T^ggo,  38°,  m°)  in  terms  of  r. 

269.  Find  the  area  of  P(i68°,  178°,  107°,  m°  so-,  177°  14-,  154°  is-)  on  a  sphere  of 
radius  1  ft. 


SECTION   IV.     VOLUME   OF   A   SPHERE  AND   ITS 
PARTS 

300.  One  method  of  finding  the  volume  of  a  sphere  is 
to    consider  it  the  limit  approached  by   a  circumscribed 
polyhedron  as  the  number  of  its  faces  is  increased  indefi- 
nitely.    This  corresponds  to  the  method  used  to  find  the 
area  of  a  circle.     As  in  dealing   with   the   circle,  there 
must  be  assumed  a  statement  that  the  figure  approaches 
a  limit  as  the  number  of  its  faces  is  increased. 

If  each  vertex  of  a  circumscribed  polyhedron  is  joined  to 
the  center  of  the  sphere,  and  planes  are  passed  tangent  to 
the  sphere  at  the  points  where  these  lines  meet  the  surface 
(thus  forming,  with  the  intercepted  parts  of  the  faces  of  the 
original  polyhedron,  a  new  polyhedron  of  a  larger  number 
of  faces),  the  area  and  the  volume  of  the  polyhedron  ap- 
proach the  area  and  the  volume  of  the  sphere  as  limits,  as 
the  number  of  faces  is  increased  indefinitely. 

301.  Theorem   XX.     The  volume  of  a  sphere  is  one 
third  the  product  of  its  radius  and  its  area. 


FIRST  METHOD.  The  volume  of  a  circumscribed  poly- 
hedron is  one  third  its  area  times  the  radius  (why?). 
Increase  the  number  of  faces  indefinitely. 

SECOND  METHOD.  Show  that  a  sphere  of  radius  r,  and 
a  cylinder  of  radius  r  and  altitude  2  r  that  has  been  hol- 
lowed out  from  each  base  to  the  center  in  the  form  of  right 
circular  cones  of  radius  r  and  altitude  r,  are  Cavalieri 

350 


VOLUME   OF   A   SPHERE   AND  ITS   PARTS 


351 


bodies,  since  the  section  of  each  at  a  distance  d  from  the 
center  is  irr2  —  Trd2.  Then  find  the  volume  of  the  hol- 
lowed cylinder. 

302.  Solids  Extending  from  the  Center  of  the  Sphere  to 
its  Surface.     The  first  method  of  proof  is  applicable  to  all 
solids  that  extend  from  the  center  of  the  sphere  to  some 
part  of  its  surface,  if  they  are  bounded  laterally  by  planes 
or  conical  surfaces,  for  all  such  solids  can  be  considered 
as  limits  of  the  sum  of  an  indefinitely  large  number  of 
pyramids  with  vertices  at  the  center  of  the  sphere,  and 
bases  tangent  to  the   sphere. 

303.  Spherical  Cones  and  Spherical  Sectors.     That  part 
of  a  sphere  included  in  a  conical  space  whose  vertex  is  at 
the  center  of  the  sphere  is  called  a  spherical  cone  (as  0—  1^). 


It  could  be  generated  by  revolving  a  plane  sector  about  one 
of  its  bounding  radii  as  an  axis,  so  it  is  considered  as  one 
kind  of  spherical  sector.  That  part  of  a  sphere  generated 
by  revolving  a  plane  sector  about  a  diameter  is  called  a 
spherical  sector  (as  0  —  J?2)>  ^  ^  revolves  about  a  diameter 
other  than  one  of  its  own  boundaries,  it  is  evidently  the 


352  POLYHEDRAL  ANGLES   AND  SPHERES 

difference  of  spherical  cones.  The  base  of  any  kind  of 
spherical  sector  is  evidently  a  zone.  The  spherical  cone 
0-Sl  is  cut  out  of  sphere  0  by  the  conical  space  0—OD. 

*  304.    The  volume  of  a   spherical* cone  or  a  spherical 
sector  is  one  third  its  base  times  the  radius  of  the  sphere. 

F=f  7rr2A 
(.where  h  is  the  altitude  of  the  zone). 

270.  Express  the  formula  for  the  volume  of  a  spherical  cone  in 
terms  of  the  distance   of   the  plane  cutting  off  the  zone   from  the 
center  of  the  sphere,  instead  of  in  terms  of  h. 

271.  Express  the  formula  for  the  volume  of  a  spherical  cone  in 
terms  of  the  radius  of  the  circle  instead  of  in  terms  of  h. 

272.  Find  the  volume  of  a  spherical  sector  in  a  sphere  of  radius 
12  in.  if  the  radii  of  the  two  circles  of  its  base  are  6  in.  and  8  in. 

305.  Spherical  Pyramids.  That  part  of  a  sphere  in- 
cluded in  a  pyramidal  space  whose  vertex  is  at  the  center 
of  the  sphere  is  called  a  spher- 
ical pyramid.  The  base  of  a 
spherical  pyramid  is  evidently 
a  spherical  polygon.  The 
spherical  pyramid,  like  any 
other  pyramid,  can  be  tri- 
angular, quadrangular,  etc. 
O-ABC  is  a  triangular  spheri- 
cal pyramid. 

*  306.    The  volume  of  a  spherical  pyramid  is  one  third 
its  base  times  the  radius  of  the  sphere. 


3     720° 
where  E  is  the  spherical  excess  of  the  base. 

273.    State  the  volume  of  a  spherical  pyramid  in  form  correspond- 
ing to  that  used  for  the  area  of  a  spherical  triangle  in  §  297. 


VOLUME  OF   A   SPHERE  AND   ITS   PARTS         353 

307.  The  Spherical  Wedge.  That  part  of  a  sphere 
between  two  great  semicircles  is  called  a  spherical  wedge 
or  an  ungula.  Its  curved  surface  is  evidently  a  lune. 

*  308.    The  volume  of  a  spherical   wedge  is  one  third 
its  base  times  the  radius  of  the  sphere. 


3     360° 
where  A  is  the  angle  between  the  circles. 

274-   State  the  volume  of  a  spherical  wedge  in  form  corresponding 
to  that  used  for  a  lune  in  §  295. 

309.  Spherical  Segments.     The  part  of  a  sphere  that 
lies  between  parallel  planes  is  called  a  spherical  segment. 
The  planes  can  both  be  secant  planes,  one  can  be  a  tan- 
gent plane,  or,  in  the  limiting  case  of  the  whole  sphere, 
both  can  be  tangent  planes.     A  spherical  segment  will 
have  one  base  or  two  bases  according  as  it  is  included 
between  a  secant  plane  and  a  tangent  plane,  or  between 
two  secant  planes.     The  spherical  surface  of  a  spherical 
segment  is  a  zone. 

310.  Volume  of  a  Spherical  Segment.    The  second  method 
of  proof  used  for  finding  the  volume  of  a  sphere  is  appli- 
cable also  to  spherical  segments,  for  the  spherical  segment 
and  the  corresponding  part  of  the  hollowed  out  cylinder 
are   Cavalieri   bodies,  and    the  volume  of   the   hollowed 
cylinder  can  be  found  by  the  formulas  for  the  cylinder 
and  the  cone.     The  volume  of  a  spherical  segment  of  one 

-in 

base  is  expressed  by  the  formula  r=^— (3r  —  ^),  and 

3 

that  of  two  bases  is  expressed  by  the  formula 


354  POLYHEDRAL  ANGLES   AND   SPHERES 

where  h  is  the  altitude,  and  /^  and  r2  are  the  radii  of  the 
bases.  These  formulas,  especially  the  second,  are  not 
easy  to  deduce,  and  are  not  necessary  for  calculation,  as 
the  Prismatoid  Formula  applies  to  spherical  segments,  and 
is  more  convenient  since  it  can  be  used  in  both  of  these 
cases  as  well  as  for  the  other  solids  already  noted. 

275.  A  sphere  is  inscribed  in  a  cylinder.     Find  the  ratios  of  their 
areas  and  of  their  volumes. 

276.  A  cylindrical  pail  of  radius  10  in.  is  full  of  water.     If  a  sphere 
of  radius  12   in.  is  set   in   the  top  of  the  pail,  how  much  water  is 
spilled,  counting  231  cu.  in.  to  the  gallon  ? 

277.  A  sphere  of   radius  10  in.  is  hollowed  out  so.  as  to  leave  a 
hollow  bounded  by  the  lateral  surface  of  the  frustum  of  a  right  cir- 
cular cone  of  radii  6  in.  and  8  in.  with  a  diameter  as  axis.     Find  the 
volume  of  the  remaining  solid. 

278.  A  sphere  of  10  in.  radius  is  hollowed  out  so  as  to  leave  a 
hollow  bounded  by  a  conical  surface  whose  vertex  is  at  one  end  of  the 
diameter  that  is  the  axis  of  the  cone.     If  the  radius  of  the  base  of  the 
cone  is  6  in.,  find  the  volume  of  the  remaining  figure. 

279.  In  building  some  decorations,  certain  trirectangular  corners 
were  filled  in  with  blocks  terminated  by  spherical  surfaces  of  radius 
3  in.  and  having  the  corners  as  centers.     What  was  the  spherical  area 
and  the  volume  of  each  block  ? 

280.  If  the  number  of  square  units  in  the  area  of  a  sphere  is  equal 
to  three  times  the  number  of   cubic  units   in  its  volume,  find  the 
radius. 

281.  A  plane  passes  through  a  sphere  so  that  it  cuts  off  one  third 
the  surface.     What  part  of  the  volume  does  it  cut  off? 

282.  The  rays  of  light  from  a  point  illumine  one  fourth  of  the  sur- 
face of  a  sphere  of  radius  4  ft.     How  far  is  the  point  from  the  sur- 
face? 

283.  The  volume  of   the  moon  is  approximately  fa  that  of   the 
earth.  What  is  the  diameter  of  the  moon  if  that  of  the  earth  is  7916 
miles  ? 


VOLUME   OF   A   SPHERE   AND   ITS   PARTS         355 

284.  What  is  the  radius  of  a  sphere  such  that  the  number  of  cubic 
units  in  its  volume  equals  the  number  of  square  units  in  the  area  of 
a  great  circle?  such  that  it  equals  the  number  of  linear  units  in  the 
circumference  of  a  great  circle  ? 

285.  What  is  the  area  of  a  sphere  whose  area  contains  the  same 
number  of  square  units  that  there  are  linear  units  in  the  circumfer- 
ence of  a  great  circle  ? 

286.  How  many  marbles  one  half  inch  in  diameter  can  be  made 
by  melting  a  sheet  of  glass  4  ft.  by  3  ft.  and  f  in.  thick  ? 

287.  Show  that  the  volume  of  a  sphere  inscribed  in  a  cube  of  edge 
E  is  approximately  .7854  Es. 

288.  Show  that  the  volume  of  a  cube  inscribed  in    a  sphere   of 
radius  r  is  approximately  1.6396  r3. 

289.  Which  is  the  better  bargain,  apples  at  20^  a  dozen,  or  apples 
one  and  a  half  times  as  great  in  diameter  at  60 1  a  dozen  ? 

290.  The  diameter  of  the  sun  is   111^  times  that   of   the  earth. 
How  does  its  volume  compare  with  that  of  the  earth  ?  its  area  ? 

291.  If  the  diameter  of  the  moon  is  2000  miles,  and  that  of  the 
earth  8000  miles,  how  far  from  the  surface  of  the  earth  would  the 
moon  have  to  be  to  shine  on  one  fourth  of  the  earth's  surface  at  a 
time? 

292.  Using  the  same  diameters  as  in  the  last  exercise,  find  on  what 
part  of  the  earth's  surface  the  moon  shines  if  the  distance  between 
the  centers  of  the  earth  and  the  moon  is  238,000  miles. 

293.  What  is  the  volume  of  a  sphere  if  a  circle  of  radius  4  in.  has 
a  polar  distance  of  45°?  of  30°?  of  60°?  of  90°? 

294-  If  a  spherical  triangle  of  angles  90°,  85°,  and  125°  has  an 
area  of  %5  TT  sq.  in.,  what  is  the  volume  of  the  spherical  pyramid  on 
that  base? 

295.  If  the  volume  of  a  spherical  wedge  of  72°  is  -3/7r,  find  the 
area  of  a  lune  of  72°  on  the  same  sphere. 

311.  The  Degree  of  Area  and  Volume  Formulas.  In  plane 
geometry,  it  has  been  found  that  the  formulas  for  the 
areas  of  all  plane  figures  are  of  second  degree  in  terms 
of  lengths  of  sects,  and  that  the  areas  of  similar  figures 
are  proportional  to  the  squares'  of  corresponding  sheets. 


356  POLYHEDRAL   ANGLES   AND   SPHERES 

Similarly,  the  formulas  for  the  areas  of  all  solids  studied 
are  of  second  degree,  either  involving  the  second  degree 
of  one  length, — as,  the  area  of  a  sphere  equals  4  ?rr2, 
or  the  product  of  two  lengths,  —  as,  the  lateral  area  of 
a  cylinder  equals  2  Trrh.  (Note  that  TT  is  a  number,  and 
so  does  not  affect  the  degree.) 

The  formulas  for  the  volumes  of  all  the  solids  studied 
are  of  third  degree,  either  involving  the  cube  of  one 
length,  —  as,  the  volume  of  a  sphere  equals  ^Trr3;  the 
square  of  one  length  times  another  length,  —  as,  the  vol- 
ume of  a  right  circular  cone  equals  -J-  7rr*h ;  or  the  pro- 
duct of  three  lengths,  —  as,  the  volume  of  a  rectangular 
parallelepiped  equals  the  product  of  its  three  dimensions. 

312.  Similar  Solids.  Polyhedrons  are  said  to  be  similar 
when  they  are  bounded  by  the  same  number  of  correspond- 
ingly similar  faces,  and  their  corresponding  polyhedral 
angles  are  equal.  It  is  evident  that  any  one  pair  of  cor- 
responding sects  would  be  proportional  to  any  other  pair 
of  corresponding  sects.  Right  circular  cones  or  cylinders 
are  called  similar  if  their  axes  are  proportional  to  the 
radii  of  their  bases.  Any  two  spheres  are  similar,  in  that 
any  pair  of  corresponding  sects  are  proportional  to  any 
other  pair  of  corresponding  sects.  The  proportionality 
of  corresponding  sects  shown  in  all  of  these  definitions 
is  the  idea  that  is  of  the  most  importance  in  elementary 
geometry.  The  definitions  are  sometimes  summed  up  by 
saying  that  any  two  solids  are  similar  if  they  can  be  placed 
so  that  any  two  corresponding  points  will  cut  a  sheaf  of 
lines  in  the  same  ratio  as  any  other  pair  of  corresponding 
points.  Show  that  solids  fulfilling  the  conditions  of  this 
definition  also  satisfy  the  conditions  of  the  other  defini- 
tions. 


SUMMARY  OF   PROPOSITIONS  357 

*  313.    The  areas  of  similar  solids  are  proportional  to 
the  squares  of  any  pair  of  corresponding  sects. 

*  314.    The  volumes  of  similar  solids  are  proportional 
to  the  cubes  of  any  pair  of  corresponding  sects. 

296.  If  two  similar  solids  have  the  area  ratio  4,  what  is  their  vol- 
ume ratio  ? 

297.  What  relation  have  the  volume  ratio,  the  area  ratio,  and  the 
line  ratio,  of  two  similar  solids? 

298.  Name  six  kinds  of  solids  such  that  any  two  of  the  same  kind 
are  similar. 

299.  If  the  volumes  of  two  similar  solids  are  3141.6  cu.  ft.  and 
25,132.8  cu.  ft.,  and  the  area  of  the  first  is  1256.64  sq.  ft.,  what  is  the 
area  of  the  second  ? 

300.  Given  a  material  sphere,  construct  the  radius  of  a  sphere 
having  twice  as  great  an  area ;  any  given  number  of  times  as  great 
an  area. 

301.  On  different  spheres,  zones  of  the  same  altitude  are  propor- 
tional to  the  radii  of  the  spheres.     In  order  that  zones  on  different 
spheres  may  fulfill  the  conditions  of  the  general  definition  of  similarity, 
what  would  have  to  be  true  of  their  altitudes  ? 

302.  Given  a  regular  tetrahedron,  construct  another  having  half 
as  great  an  area.     What  is  true  of  their  volumes? 

315.  SUMMARY  OF  PROPOSITIONS 

I.   DETERMINATION: 

of  a  sphere,  by  four  points  (194)  ; 

of  a  great  circle,  by  the  center  and  two  points  on  the  sur- 
face (211); 
of  a  small  circle,  by  three  points  on  the  surface  (211). 

II.   RELATIVE  POSITIONS  ;  TANGENTS  AND  SECANTS  : 

(1)  A  point  and  a  sphere: 

The  point  is  within,  on  the  surface,  or  outside  of  a  sphere, 
according  as  its  distance  from  the  center  is  less  than, 
equal  to,  or  greater  than  a  radius  (193). 


358  POLYHEDRAL   ANGLES   AND   SPHERES 

(2)  A  line  and  a  sphere: 

The  line  is  a  secant,  a  tangent,  or  does  not  meet  the  sphere, 
according  as  its  distance  from  the  center  is  less  than, 
equal  to,  or  greater  than  a  radius  (196). 

(3)  A  plane  and  a  sphere :     ' 

The  plane  cuts  the  sphere  in  a  circle  (203),  is  tangent  to  it 
(198),  or  does  riot  meet  it  (197),  according  as  its  distance 
from  the  center  is  less  than,  equal  to,  or  greater  than,  a 
radius,  and  conversely. 

The  perpendicular  to  a  tangent  plane  at  its  point  of  con- 
tact passes  through  the  center  of  the  sphere  (200). 

(4)  Two  spheres 

are  tangent  if  the  center  sect  equals  the  sum  of  their  radii 
(or  if  they  meet  on  the  line  of  centers)  (234)  ; 

are  tangent  if  they  are  tangent  to  the  same  plane  at  the 
same  point  (235)  ; 

intersect  in  a  circle  if  they  meet  at  a  point  not  on  the  line 
of  centers  (236). 

III.   CIRCLES  OF  A  SPHERE: 

(1)    The  great  circle  : 

Its  center  is  the  center  of  the  sphere  (208). 

It  is  the  largest  circle  of  a  sphere  (206). 

Two  great  circles  on  the  same  sphere  intersect  in  a  di- 
ameter (209). 

The  great  circle  bisects  the  sphere  and  its  surface  (210). 

The  great  circle  is  determined  by  the  center  and  two  points 
on  the  surface  (211). 

The  great  circle  is  perpendicular  to  a  circle  if  it  contains 
one  of  its  poles,  and  conversely  (213). 

Its  arc  is  perpendicular  to  a  circumference  of  a  great  circle 
if  it  contains  one  of  its  poles  (224). 

Its  polar  distance  is  a  quadrant,  and  conversely  (218). 

The  shortest  line  on  the  surface  between  two  points  is  the 
minor  great  circle  arc  (247). 


SUMMARY   OF   PROPOSITIONS  359 

(2)  Axes  and  poles : 

The  line  perpendicular  to  a  circle  at  its  center,  or  joining 

its  center  to  the  center  of  the  sphere,  is  its  axis  (205). 
Parallel  circles  have  the  same  axis  and  poles  (212). 
A  pole  is  equidistant  from  all  points  on  the  circumference 

(214). 
Polar  distances  of  the  same  or  of  equal  circles  are  equal 

(216). 
Either  of  two  points  a  quadrant's  distance  apart  is  the  pole 

of  a  great  circle  through  the  other  (219). 
A  point  at  a  quadrant's  distance  from  each  of  two  other 

points  is  the  pole  of  a  great  circle  through  them  (220). 
A  polar  chord  is  the  mean  between  the  diameter  and  its 

own  projection  upon  the  axis  (229). 
The  radius  of  a  circle  is  the  mean  between  the  sects  of  its 

axis  (229). 

(3)  Miscellaneous: 

Circles  equidistant  from  the  center  of  a  sphere  are  equal, 
and  conversely  (206). 

Circles  not  equidistant  from  the  center  are  unequal,  the 
one  farther  from  the  center  being  smaller,  and  con- 
versely (206). 

If  two  circumferences  meet  at  a  point  on  the  great  circle 
circumference  through  their  poles,  they  cannot  meet 
again  (246). 

IV.   PROPERTIES  OF  THE  SPHERE  : 

Radii  or  diameters  of  the  same  or  of  equal  spheres  are 
equal  (190,  191). 

Spheres  of  equal  radii  are  congruent  (191). 

One,  and  but  one,  sphere  can  be  circumscribed  about  a 
tetrahedron  (195). 

One,  and  but  one,  sphere  can  be  inscribed  in  a  tetra- 
hedron (202). 

Its  surface  is  the  locus  of  points  at  a  fixed  distance  from 
a  given  point  (its  center)  (192). 


360  POLYHEDRAL   ANGLES   AND   SPHERES 

V.   MEASUREMENT  OP  A  SPHERICAL  ANGLE: 

by  the  measuring  angle  of  the  dihedral  angle  between  the 

planes  in  which  its  arms  lie  (223)  ; 
by  the  subtended  part  of  the  great  circle  circumference  of 

which  it  is  a  pole  (226)  ; 
when  it  is  an  angle  of  a  triangle,  by  the  supplement  of  the 

opposite  side  of  the  polar  triangle  (275). 

VI.   POLYHEDRAL  ANGLES  AND  SPHERICAL  POLYGONS  : 

(1)  Relations  between  polyhedral  angles  and  spherical  polygons : 

The  face  angles  of  a  central  polyhedral  angle  are  measured 
by  the  subtended  sides  of  its  spherical  polygon  (242). 

The  dihedral  angles  of  a  central  polyhedral  angle  are 
measured  by  the  same  measuring  angle  as  the  corre- 
sponding angles  of  its  spherical  polygon  (243). 

(2)  Properties  of  a  spherical  polygon  (polyhedral  angle). 

The  sum  of  the  sides  (face  angles)  of  a  spherical  polygon 

(polyhedral  angle)  is  less  than  a  circumference  (peri- 

gonj  (249). 
In  a  spherical  triangle  (trihedral  angle)  the  sum  of  the 

spherical  angles  (dihedral  angles)  is  more  than  one,  and 

less  than  three,  straight  angles  (277). 
In  a  spherical  triangle  the  sum  of  two  sides  (face  angles) 

is  greater  than  the  third  (245). 
In  an  isosceles  spherical  triangle  (trihedral  angle)  the  base 

angles  (opposite  dihedral  angles)  are  equal  (268). 
In  an  isosceles  spherical  triangle  the  bisector  of  the  vertex 

angle,  the  median  to  the  base,  the  altitude  to  the  base, 

and  the  perpendicular  bisector  of  the  base,  are  all  the 

same  line  (269). 

(3)  Congruence  and  symmetry  : 

Vertical  spherical  polygons  (polyhedral  angles)  are  sym- 
metric (261,  262). 

Spherical  triangles  (trihedral  angles)  are  congruent  or 
symmetric  if  the  following  parts  of  one  are  equal  to  the 
corresponding  parts  of  the  other  : 


SUMMARY  OF   PROPOSITIONS  361 

(a)  two  sides  and  the  included  angle  (two  face  angles  and 
the  included  dihedral  angle)  (266)  ; 

(&)  two  angles  and  the  included  side  (two  dihedral  angles, 
and  the  included  face  angle)  (267)  ; 

(c)  three  sides  (three  face  angles)  (271)  ; 

(d)  three  angles  (three  dihedral  angles)  (276)  ; 
Isosceles  symmetric  spherical  triangles  (trihedral  angles) 

are  congruent  (270). 
Symmetric  spherical  triangles  are  equivalent  (272). 

(4)  Polar  triangles : 

If  one  triangle  is  the  polar  of  a  second,  the  second  is  also 
the  polar  of  the  first  (274). 

An  angle  of  one  of  the  two  polar  triangles  and  the  op- 
posite side  of  the  other  have  supplemental  measures 
(275). 

VII.   Locus  OF  POINTS  : 

The  locus  of  points  at  a  fixed  distance  from  a  given  point 
is  the  spherical  surface  with  the  point  as  center  and  the 
distance  as  radius  (192). 

The  locus  of  points  in  a  sphere  equidistant  from  all  points 
on  the  circumference  of  a  circle  is  its  axis  (214). 

VIII.   THE  MATERIAL  SPHERE  : 

To  describe  a  circumference  with  a  given  pole  and  a  given 

polar  chord  (228). 

To  find  the  radius  of  a  given  circumference  (227). 
To  find  the  diameter  of  a  given  sphere  (230). 

IX.  LIMITING  CASES  : 

The  zone  is  the  limit  approached  by  the  area  generated  by 
a  broken  line  of  equal  sects  inscribed  in  the  generating 
arc  of  the  zone  (282). 

The  area  and  the  volume  of  a  sphere  are  the  limits  ap- 
proached by  the  area  and  the  volume  of  a  circumscribed 
polyhedron  (300). 


362  POLYHEDRAL   ANGLES   AND   SPHERES 

X.    REGULAR  POLYHEDRONS  AND  SIMILAR  SOLIDS  : 

There  can  be  but  five  regular  polyhedrons  (251). 

The  areas  of  similar  solids  are  proportional  to  the  squares 

of  corresponding  sects  (313). 
The  volumes  of  similar  solids  are  proportional  to  the  cubes 

of  corresponding  sects  (314). 

XL    LUNES  : 

Limes  are  equal  if  their  angles  are  equal,  and  conversely 

(290,  291). 
Adding  or  subtracting  lunes  adds  or  subtracts  their  angles 

(292). 
Multiplying  a  lime  multiplies  its  angle  by  the  same  factor 

(293). 

Lunes  are  proportional  to  their  angles  (294). 
A  lune  is  the  same  part  of  a  spherical  surface   that   its 

angle  is  of  a  perigon,  or  360°  (295). 

XII.  AREA  ON  A  SPHERE: 

(1)  of  the  sphere  =  4  Trr2  (284)  ; 

(2)  of  a  zone  =  2  -nrh  (283)  ; 

(3)  of  a  lune  =  r^  (4irr2),  where  A  is  its  angle  in  degrees  (295) ; 

(4)  of  a  spherical  triangle  =  that  of  a  lune  whose  angle  is  one 

half  its  spherical  excess,  or  -^  (4  Trr2)  (296,  297)  ; 

(5)  of  a  spherical  polygon  =  -^  (4  Trr2)  (298). 

XIII.  VOLUME  OP  A  SPHERE  AND  ITS  PARTS: 

(1)  of  the  sphere  =  |  area  x  radius  =  f  Trr8  (301)  ; 

(2)  of  the  spherical  cone  (304),  spherical  sector  (304),  spherical 

pyramid  (306),  spherical  wedge  (308),  =  £  radius  x  area 
of  its  spherical  base  =  ^  br ; 

(3)  of  a  spherical  segment.     Use  the  Prismatoid  Formula  (310). 

316.  ORAL  AND  REVIEW  QUESTIONS 

State  the  theorems  in  this  book  about  spherical  triangles  that 
correspond  to  theorems  about  plane  triangles.  What  important 
difference  is  there  between  the  angles  of  a  plane  triangle  and  those 


ORAL   AND   REVIEW   QUESTIONS  363 

of  a  spherical  triangle?  Why  is  this  so?  How  many  points  deter- 
mine a  sphere  ?  Why?  How  many  points  013.  the  surface  determine 
a  great  circle?  Why?  a  small  circle?  Why  are  circles  through 
the  center  called  great  circles  ?  Why  are  circles  equidistant  from  the 
center  equal?  Define  axis,  pole,  polar  distance.  What  is  the  polar 
distance  of  a  great  circle  ?  In  what  form  is  the  converse  of  this  fact 
used  ?  How  is  the  diameter  of  a  material  sphere  found  ?  In  a  sphere 
of  radius  10  in.,  how  far  from  the  center  is  a  circle  of  6  in.  radius? 
What  is  its  polar  chord?  Which  methods  of  proving  triangles  con- 
gruent or  symmetric  use  superposition  ?  What  are  the  other  methods? 
Can  one  triangle  be  polar  to  a  second,  and  the  second  be  polar  to  a 
third?  Explain  fully  which  poles  are  taken  in  forming  a  polar  tri- 
angle. What  numerical  relation  holds  between  parts  of  two  polar 
triangles?  If  the  three  sides  of  one  of  two  polar  triangles  are  97°, 
83°,  69°,  find  as  many  parts  of  the  other  triangle  as  possible. 

How  is  the  formula  for  the  area  of  a  sphere  obtained?  What  part 
of  the  surface  has  its  formula  derived  in  the  same  way  ?  How  is  the 
area  of  a  lune  found  ?  of  a  spherical  triangle  ?  of  a  spherical  polygon  ? 
What  group  of  solids  has  its  volumes  expressed  by  the  formula 

-^,  and  for  what  does  b  stand  in  each  case  ?  What  other  parts  of  a 
3 

sphere  are  there,  and  how  are  their  volumes  found?  What  is  the 
intersection  of  a  sphere  and  a  plane?  Why?  What  is  the  intersec- 
tion of  two  spheres?  Why?  When  is  a  plane  tangent  to  a  sphere? 
When  are  two  spheres  tangent  to  each  other  ? 

Of  what  is  a  spherical  surface  the  locus?  What  is  the  locus  of 
points  at  a  fixed  distance  from  a  point,  and  at  the  same  time  equi- 
distant from  that  point  and  another  point?  at  a  fixed  distance  from 
each  of  two  given  points?  What  is  the  only  distinction  between  con- 
gruence and  symmetry  of  spherical  triangles?  If  a  great  circle  is 
perpendicular  to  a  small  circle,  what  must  it  contain  ?  Is  this  suffi- 
cient to  make  it  a  perpendicular?  What  is  known  about  the  sum  of 
the  face  angles  of  a  polyhedral  angle  ?  What  spherical  theorem  fol- 
lows from  this?  What  is  known  about  the  face  angles  of  a  trihedral 
angle  ?  What  spherical  theorem  follows  ?  What  are  the  limits  of  the 
sum  of  the  sides  of  a  spherical  polygon  ?  of  the  sum  of  the  angles  of  a 
spherical  triangle?  How  many  equilateral  triangles  can  be  used  to 
form  a  solid  angle?  Why?  What  are  the  regular  polyhedrons  and 
why  are  they  the  only  ones  ?  State  the  area  and  volume  formulas  for 
the  sphere  and  all  its  parts.  How  many  of  the  volume  formulas  can 


364  POLYHEDRAL   ANGLES  AND   SPHERES 

be  derived  from  the  Prismatoid  Formula?  Which  could  be  proved 
by  the  use  of  Cavalierfs  Theorem,  and  how?  Upon  what  power  of 
the  lengths  of  sects  do  area  formulas  depend  ?  volume  formulas  ?  If 
the  solids  are  similar,  what  rule  holds  for  the  area  and  the  volume  ? 
If  the  line  ratio  is  3,  what  are  the  area  and  volume  ratios?  If  the 
volume  ratio  is  3,  what  are  the  line  and  area  ratios?  If  the  area  ratio 
is  3,  what  are  the  line  and  volume  ratios  ?  State  all  the  ways  in  which 
spherical  angles  can  be  measured.  What  relation  is  there  between 
polyhedral  angles  and  spherical  polygons?  Explain  why.  How  is 
this  relation  used  ?  What  is  meant  by  "  distance  "  on  a  spherical  sur- 
face, and  why  ?  Explain  how  a  spherical  triangle  can  have  three  right 
angles,  and  tell  what  its  polar  triangle  is.  How  are  symmetric  tri- 
angles proved  equivalent? 

GENERAL  EXERCISES 

803.  If  the  strength  of  material  is  in  proportion  to  its  cross  section, 
what  is  the  effect  of  doubling  the  diameter  of  a  wire  ?  If  a  wire  of 
|  in.  radius  will  support  an  iron  ball  of  radius  2  in.,  how  large  a  wire 
will  support  a  ball  of  radius  4  in.?  What  effect  has  doubling  the 
diameter  of  a  tree  on  the  amount  of  wood  in  it  ? 

304-  How  many  tangents  can  be  drawn  from  an  external  point  to 
a  sphere  ?  How  could  one  .be  constructed  ?  How  could  a  plane  be 
drawn  from  an  external  point  tangent  to  a  sphere  ? 

305.  Find  the  locus  of  the  centers  of  all  spheres  that  are  tangent  to 
a  given  plane  at  a  given  point. 

306.  Find  the  locus  of  the  centers  of  all  spheres  of  given  radius 
that  are  tangent  to  a  given  plane. 

307.  Find  the  locus  of  the  centers  of  all  spheres  that  are  tangent 
to  the  faces  of  a  trihedral  angle. 

308.  A  cylindrical  hole  of  radius  6  in.  was  bored  through  a  sphere 
of  radius  10  in.,  the  axis  of  the  cylinder  being  a  diameter  of  the 
sphere.     What  was  the  weight  of  the  remaining  part  of  the  sphere, 
if  its  entire  weight  was  35  Ib.  ?     What  ratio  had  the  total  area  of  the 
remaining  figure  to  the  area  of  the  sphere  ? 

309.  To  inscribe  a  cube  in  a  given  sphere. 

310.  To  inscribe  a  regular  tetrahedron  in  a  given  sphere. 

311.  To  inscribe  a  regular  octahedron  in  a  given  sphere. 


GENERAL  EXERCISES  365 

812.    How  large  an  iron  shell  1  in.  thick  would  a  sphere  of  radius 
3  in.  make  if  melted  and  recast  ? 

313.  A  cup  is  in  the  form  of  a  segment  of  a  sphere  of  internal 
radius  3  in.,  with  a  flat  base  of  radius  2  in.,  and  a  cylindrical  top  of 
radius  2J  in.,  and  height  2  in.  How  much  water  will  it  hold  ?  (231 
cu.  in.  =  1  gal.) 

314.  Why  is  a  plumb  line,  in  general,  perpendicular  to  the  surface 
of  the  earth  ? 

315.  In  mixing  paint,  it  is  important  to  know  of  what  sizes  to  grind 
the  spherical  particles  of  coloring  matter  in  order  to  cover  the  surface 
evenly.     If  spheres  of  radius  r  are  tangent  to  each  other  and  to  a 
plane,  find  what  size  other  spheres  must  be  so  that  they  can  lie  between 
each  three  of  those  spheres  and  be  tangent  to  them  and  to  the  plane. 

316.  If  spheres  of  radius  r  are  piled  up  in  pyramidal  form  so  that 
each  one  rests  on  three  others  that  are  tangent  to  one  another,  show 
that  the  centers  of  the  outside  spheres  lie  in  the  faces  of  a  regular 
tetrahedron.     Find  the  height  from  the  highest  point  to  the  ground 
if  cannon  balls  of  radius  6  in.  are  piled  three  deep. 

*•  317.  If  the  edge  of  a  cube  is  doubled,  what  is  the  effect  on  its  area? 
on  its  volume  ?  If  the  area  is  doubled,  what  is  the  effect  on  its  edge  ? 
on  its  volume?  If  its  volume  is  doubled,  what  is  the  effect  on  its 
edge?  on  its  area? 

318.  Prove  that  the  shortest  sect  from  a  given  point  to  the  surface 
of  a  given  sphere  is  along  a  line  to  the  center.     (Two  cases.) 

319.  Circles  are  inscribed  in,  and  circumscribed  about,  an  equi- 
lateral triangle.     Find  the  ratios  of  the  areas  and  the  volumes  of  the 
solids  generated  by  revolving  the  triangle  and  the  circles  about  an 
altitude  of  the  triangle  as  an  axis. 

320.  If  a  right  circular  cylinder  of  altitude  equal  to  its  diameter 
and  a  right  circular  cone  whose  slant  height  equals  its  diameter  are 
circumscribed  about  a  sphere,  show  that  the  total  area  of  the  cylinder 
is  the  mean  proportional  between  the  areas  of  the  cone  and  the  sphere, 
and  that  the  volume  of  the  cylinder  is  the  mean  proportional  between 
the  volumes  of  the  cone  and  the  sphere. 

321.  How  many  square  inches  of  gold  leaf  are  required  to  gild  a 
sphere  6  ft.  in  diameter,  no  allowance  being  made  for  waste? 


366  POLYHEDRAL   ANGLES  AND  SPHERES 

322.  A  manufacturer  of  marbles  uses  a  sheet  of  white  glass  3  ft. 
square  and  1  in.  thick,  a  sheet  of  green  glass  in  the  form  of  a  cylin- 
drical surface  of  element  2  ft.,  thickness  f  in.,  and  length  along  a 
line  perpendicular  to   the   elements  28  in.,  and    a   piece  of  yellow 
glass  which  when  placed  in  a  tank  3  ft.  deep  that  is  a  circular  frustum 
of  radii  2  ft.  and  3  ft.,  raises  the  water  from  1  ft.  deep  to  2  ft.  3  in. 
deep.     How  many  marbles  f  in.  in  diameter  can  be  made  by  mixing 
these  glasses,  and  what  proportion  of  each  kind  of  glass  will  be  in 
each? 

323.  If  all  possible  lines  are  drawn  from  a  point  to  a  sphere,  the 
product  of  two  sects  from  .the  vertex  to  the  surface  on  any  one  secant 
equals  the  product  of  the  sects  from  the  vertex  to  the  surface  on  any 
other  secant,  and  any  tangent  is  the  mean  proportional  between  the 
sects  of  any  secant. 

324'   To  construct  a  sphere  of  given  radius  tangent  to  a  given 
plane  at  a  given  point. 

325.  Find  the  locus  of  a  point  such  that  the  ratio  of  its  distances 
from  two  given  points  is  constant. 

326.  In  filling  a  measure  with  fruit  or  vegetables  of  approximately 
spherical  shape,  would  a  larger  quantity  be  obtained  with  those  of 
lesser  radius,  or  with  those  of  greater  radius  ?     Why  ? 

327.  If  cannon  balls  of  1  ft.  diameter  are  piled  four  deep  so  that 
each  one  rests  on  four  others  that  are  tangent  to  each  other,  find  the 
height  of  the  pile. 

328.  A  cylindrical  fire  extinguisher  is  8  in.  in  diameter  and  2  ft. 
long  with  a  spherical  segment  3  in.  high  on  its  end  as  a  great  circle. 
How  much  liquid  is  required  to  charge  it?     How  long  will  it  take  to 
empty  it  by  a  hose  \  in.  in  diameter  if  the  liquid  flows  through  the 
hose  at  the  rate  of  25  ft.  per  second  ? 


GENERAL   SUMMARY 

OF    THE       • 

FORMULAS  OF  SOLID  GEOMETRY 

317.     MEANING  OP  THE  LETTERS  USED  IN  THE  FORMULAS 

h  =  altitude,  the  perpendicular  from  the  vertex  to  the  base,  or  between 

the  two  bases. 
s  =  the  slant  height  in  a  regular  pyramid  or  its  frustum,  the  element 

in  a  cylinder,  and  in  a  right  circular  cone  or  its  frustum,  the  edge  in 

a  prism, 
ft,  />,  r :   when  a  figure  has  but  one  base,  b  is  its  area,  p  is  its  perimeter, 

and,  if  it  is  circular,  r  is  its  radius. 
&!  and  62,  p1  and  p2,  r^  and  r2,  m  :   when  a  figure  has  parallel  bases  ^ 

and  2>2  are  their  areas,  pl  and  p2  are  their  perimeters,  and,  if  they  are 

circular,  rt  and  rz  are  their  radii ;  m  is  the  area  of  the  midsection. 
prt  and  rrt  are  the  perimeter  and  radius  of  a  right  section. 

I.  AREA. 

(1)  Lateral  Area. 

(a)  Frustum  of  a  pyramid  =  g(-Pi  +  Pv  (derived  from  trap- 

ezoid). 

Applies  to  the  frustum  of  a  regular  pyramid,  the  entire 
pyramid  (/>2  =  0),  the  frustum  of  a  right  circular  cone, 
the  entire  cone  (jo2  =  0),  the  right  prism  and  the  right 
circular  cylinder  (pl  =  jt?2) 

For  figures  with  circular  bases,  it  becomes  Trs(rl  +  ra.) 
(&)  Pyramid  =  ~£  (derived  from  the  triangle). 

Applies  to  the  regular  pyramid  and  the  right  circular 

cone. 

For  the  cone  it  becomes  irrs. 
(c)  Prism  =  sprt  (derived  from  the  parallelogram). 

Applies    to    the    prism    (including  parallelepiped),  and 

cylinder. 

For  the  cylinder  it  becomes  2  Trrrts. 
307 


3G8  GENERAL   SUMMARY   OF   FORMULAS 

(2)  Area  on  a  Sphere. 

(a)  Sphere  =  4  Trr2  (rotating  a  semicircle  about  its  diameter). 
(6)   Zone  =  2  Trrh  (rotating  an  arc  about  its  diameter). 

(c)  Lune  =  -  (4  Trr2)    where  A   is   the   angle  of  the  lune 

360U 
(derived  from  its  ratio  to  the  surface  of  the  sphere). 

(d)  Spherical  polygon  =  -  -  (4  Tr2r)  where  E  is  the  spher- 

ical excess  of  the  polygon  (derived  from  the  fact  that 
a  spherical  triangle  is  equivalent  to  half  the  lune  of  its 
spherical  excess). 
II.   VOLUME. 

(1)  Prismatoid  =  —  (^  +  62  +  4  ro) 

Applies  to  any  solid  whose  bases  are  in  parallel  planes,  if  it 
is  bounded  laterally  by  planes,  or  by  curved  surfaces  gen- 
erated by  revolving  a  straight  line  sect  or  an  arc  of  a  circle 
about  an  axis. 

Applies  to  the  prismatoid,  prism,  cylinder,  pyramid  (b2  =  0), 
cone  (62  =  0),  frustum  of  a  pyramid,  frustum  of  a  cone, 
sphere  (ftx  =  62  =  0),  spherical  segment,  spherical  wedge 

(&i  =  &,  =  0). 

Does  not  apply  to  spherical  cone,  spherical  sector,  spherical 
pyramid. 

(2)  Frustum  of  a  Pyramid  or  a  Cone  =  ^(bl  +  b2  +  V6^) 

3 

(derived  from  the  difference  of  two  pyramids). 
Applies   to   the   frustum  of  a  pyramid,  the  entire  pyramid 
(bz  =  0),  the  frustum  of  a  circular  cone,  the  entire  cone 
(&2  =  0),  a  prism  and  a  circular  cylinder  (bl  —  J2). 

For  circular  figures  it  becomes  -^—  (r^  +  r22  +  ry2)' 

o 

(3)  Pyramid  or  Cone  =  — 

o 

(a  triangular  prism  is  composed  of  three  equivalent  tri- 
angular pyramids). 


For  the  cone  it  becomes  ~    . 
3 

(4)  Prism  or  Cylinder  =  bh 

(derived  from  the  rectangular  parallelepiped). 
For  the  cylinder  it  becomes 


GENERAL   SUMMARY   OF   FORMULAS  369 


(5)   Sphere  and  its  Parts. 

All  parts  from  the  center  to  the  surface,  bounded  laterally  by 

planes  or  conical  surfaces,  =  —    where   b   is   the   spherical 

3 

surface  and  r  is  the  radius  of  the  sphere 

(derived  by  taking  the  limit  approached  by  pyramids 
from  the  center  to  the  faces  of  a  circumscribed  poly- 
hedron). 

Applies  to  the  sphere,  spherical  cone,  spherical  sector,  spher- 
ical pyramid,  spherical  wedge. 

(a)  sphere  =  f  Trr3  ; 

(b)  spherical  cone  or  sector  =  f  -rrr^h  ; 

77/1        \ 

(c)  spherical  pyramid  =  -  —  -  f  -  irrs  J  ,  where  E  is  the  spher- 

ical excess  of  the  base  ; 

(</)  spherical  wedge  =  —  —  (-Trr3),  where  A   is  the  angle 
obO    \'j        / 

between  the  bounding  semicircles. 
Does  not  apply  to  spherical  segments. 


(6)  Spherical  Segment.     Use  the  Prismatoid  Formula  ;  or 

(a)  Of  one  base  =  —  (3  r  -  h  ). 

3 

(b)  Of  two  bases  =  ^  (3  r^  +  3  r22  +  £2). 


COLLEGE   EXAMINATION   QUESTIONS 

(Chosen  from  recent  examinations  of  the  College  Entrance  Examination 
Board,  the  New  York  State  Regents,  and  various  colleges.) 

829.  A  cross  section  of  a  tunnel  is  a  rectangle  8  ft.  by  12  ft.,  sur- 
mounted by  a  semicircle  whose  diameter  is  the  smaller  dimension  of 
the  rectangle.     If  the  tunnel  is  three  fourths  of  a  mile  long,  how  many 
cubic  yards  of  earth  were  removed  ?     Take  TT  to  two  decimal  places. 

830.  How  many  cubic  feet  of  masonry  will  it  take  to  build  a  dome 
in  the  form  of  a  hemisphere,  whose  outer  diameter  is  21  ft.,  and  whose 
thickness  is  15  in.? 

331.  How  should  you  find  the  center  of  a  sphere  that  is  to  be  in- 
scribed in  the  regular  tetrahedron  ABCD1 

332.  What  is  the  weight  of  a  brick  cupola  in  the  form  of  a  hemi- 
spherical shell,  6  in.  thick  and  15  ft.  in  outer  diameter,  if  the  weight 
per  cubic  ft.  is  50  Ib.  ? 

333.  A  lateral  edge  of  a  regular  pyramid  measures  12  in.,  and  each 
edge  of  its  square  base  measures  4  in.     Find  its  total  surface. 

334.  A  straight  line  perpendicular  to  one  of  two  parallel  planes  is 
perpendicular  to  the  other  also. 

335.  A  right  circular  cylinder  6  ft.  in  diameter  is  equivalent  to  a 
right  circular  cone  whose  base  is  7  ft.  in  diameter.     If  the  height  of 
the  cone  is  8  ft.,  what  is  the  height  of  the  cylinder? 

336.  A  sphere  can  be  circumscribed  about  any  given  tetrahedron. 

,837.   Define :  dihedral  angle,    projection  of  a   line  upon  a  plane, 
symmetric  trihedral  angles,  prism,  right  cylinder,  sphere. 

338.   The  volume  of  a  sphere  is  2929£  TT  cu.  in. ;  find  its  surface. 

v  339.  Find  the  ratio  of  the  volumes  of  two  similar  tetrahedrons 
whose  homologous  edges  are  as  1 : 8.  Find  the  ratio  of  their  edges  if 
their  volumes  are  as  1:8. 

340.   Find  the  locus  of  a  point  on  a  sphere  that  is  equidistant  from 
two  given  points. 

370 


COLLEGE   EXAMINATION   QUESTIONS  871 

341'  A  sphere  of  radius  10  in.  is  cut  into  two  parts  by  a  plane 
whose  distance  from  the  center  of  the  sphere  is  6  in.  Find  correct 
to  three  significant  figures  (a)  the  volume  of  the  sphere,  (/;)  the 
volume  of  the  smaller  segment. 

34^.  The  area  of  the  base  of  a  cone  of  revolution  is  &,  and  the  area 
of  the  curved  surface  is  c.  Compute  the  volume  of  the  cone. 

t/  343.   Prove  that  two  similar  tetrahedrons  are  to  each  other  as  the 
cubes  of  any  two  homologous  edges. 

344'  The  area  of  the  base  of  a  right  prism  is  12  sq.  in. ;  its  total 
area  is  295  sq.  in. ;  the  base  is  a  regular  hexagon.  Find  its  volume. 

345.  Find  the  ratio  of  the  edge  of  a  cube  to  the  edge  of  a  regular 
tetrahedron  having  the  same  volume. 

346.  If  the  area  of  a  spherical  surface  is  100  sq.  ft.,  what  is  the 
area  of  a  spherical  triangle  whose  angles  are  30°,  120°,  and  150°? 

347.  A  cone  of  revolution  and  a  cylinder  of  revolution  each  have 
as  base  a  great  circle  of  radius  r,  and  as  altitude  the  radius  of  the 
sphere.     Find  the  ratios  of  the  total  surfaces  of  the  cone  and  cylinder 
to  the  surface  of  the  sphere. 

348.  If  a  plane  is  perpendicular  to  one  of  two  parallel  lines,  it  is 
perpendicular  to  the  other. 

349.  The  plane  passed  through  two  diagonally  opposite  edges  of  a 
parallelepiped  divides  it  into  two  equivalent  triangular  prisms.     If 
the   parallelepiped   had  been  a  right  parallelepiped,   what  simpler 
method  of  proof  could  have  been  used  ? 

350.  The  intersection  of  two  planes  tangent  to  a  circular  cylinder 
is  parallel  to  the  elements  of  the  cylinder. 

351.  The  inside  of  a  glass  is  in  the  form  of  a  cone  whose  vertical 
angle  is  60°,  and  whose  base  is  2  in.  across.     The  glass  is  filled  with 
water,  and  the  largest  sphere  that  can  be  immersed  is  placed  in  the 
glass.     How  much  water  remains  in  the  glass  ? 

,352.  Find  the  altitude  of  a  frustum  of  a  circular  cone,  if  its  volume 
equals  190  cu.  cm.,  and  the  radii  of  its  bases  are  respectively  2  cm. 
and  3  cm. 

353.  Find  the  volume  of  a  pyramid  whose  base  contains  30  sq.  in., 
if  one  lateral  edge  is  5  in.,  and  the  angle  formed  by  this  edge  and  the 
plane  of  the  base  is  45°. 


372  COLLEGE   EXAMINATION   QUESTIONS 

354.  A  solid  metal  sphere  whose  radius  is  12  in.  is  recast  into  a 
spheric  shell ;  the  cavity  is- spheric  and  has  the  same  radius  as  that  of 
the  original  sphere.  Find  the  thickness  of  the  shell. 

855.  Prove  that  any  section  of  a  tetrahedron  made  by  a  plane 
parallel  to  two  opposite  edges  is  a  parallelogram. 

356.  Prove  that  the  volume  of  any  regular  pyramid  is  equal  to  one 
third  its  lateral  area  multiplied  by  the  perpendicular  distance  from 
the  center  of  the  base  to  any  lateral  face. 

357.  Prove  that  if  the  four  sides  of  a  spheric  quadrilateral  are 
equal,  its  diagonals  bisect  each  other. 

358.  What  is  the  locus  of  a  point  in  space  equally  distant  from 
three  points?     Demonstrate. 

359.  What  is  the  locus  of  a  point  which  is  equidistant  from  three 
planes  which  meet  in  three  parallel  lines?     Prove  your  answer. 

360.  If  the  four  diagonals  of  a  prism,  whose  base  is  a  quadrilateral, 
meet  in  a  point,  prove  that  the  prism  is  a  parallelepiped. 

361.  The  corner  of  a  cube  is  cut  off  by  a  plane  passing  through 
the  midpoints  of  the  edges  which  terminate  at  that  vertex,  and  the 
process  is  repeated  for  each  corner  of  the  cube.     Prove  that  the  vol- 
ume of  the  solid  that  remains  is  five  sixths  of  the  volume  of  the  cube. 

362.  Find  the  locus  of  a  point  whose  distance  from  a  fixed  straight 
line  is  equal  to  its  distance  from  -a  fixed  plane  perpendicular  to  that 
line. 

363.  Show  that  the  area  of  a  spherical  triangle  drawn  on  a  sphere 
whose  radius  is  10  in.,  is  (200  TT  —  270)  sq.  in.,  if  the  lengths  of  the 
sides  of  its  polar  triangle  are  respectively  8,  9,  and  10  in. 

364.  Find  the  locus  of  a  point  equidistant  from  the  three  faces  of 
a  trihedral  angle.     Give  proof. 

365.  Find  the  area  and  the  volume  of  a  solid  generated  by  the 
revolution  of  an  equilateral  triangle  about  one  of  its  sides  as  an  axis, 
each  side  of  the  triangle  being  4  in. 

366.  A  rectangle  6  ft.  by  4  ft.  is  revolved  about  each  of  two  of  its 
adjacent  sides.     Find  the  volume  and  the  total  surface  of  each  solid 
generated. 

367.  The  total  surface  of  a  right  cylinder  is  S  and  the  altitude  is 
equal  to  the  diameter  of  the  base.     Find,  in  terms  of  S,  the  volume  of 
the  cylinder. 


COLLEGE   EXAMINATION   QUESTIONS  373 

/••  368.  A  regular  hexagonal  pyramid  whose  altitude  is  8  inches,  and 
whose  base  is  54  V3  sq.  in.,  is  cut  by  a  plane  parallel  to  the  base ;  the 
area  of  the  section  is  £  the  area  of  the  base.  Find  (a)  the  distance  of 
the  section  from  the  base ;  (6)  a  side  of  the  section. 

869.  An  oblique  prism  is  equivalent  to  a  right  prism  whose  base 
is  a  right  section  of  the  oblique  prism,  and  whose  altitude  is  equal  to 
a  lateral  edge  of  the  oblique  prism. 

370.  Find  the  surface  of  a  lune  whose  angle  is  30°,  on  a  sphere 
whose  radius  is  5  ft. 

371.  If  the  altitude  of  a  cone  of  revolution  is  equal  to  the  diameter 
of  its  base,  find  the  altitude  if  the  volume  equals  that  of  a  given 
sphere  whose  diameter  is  d. 

372.  An  equilateral  triangle  is  revolved  about  one  of  its  altitudes 
as  an  axis.    If  a  side  of  the  triangle  is  18,  find  the  volume  of  the  cone 
thus  generated,  and  the  area  of  the  surface  of  the  sphere  inscribed 
in  it. 

373.  Find  the  volume  of  a  regular  hexagonal  pyramid  the  sides  of 
whose  bases   are  16  and  8    respectively,  and  whose  lateral   edge  is 
8V10. 

374-  (a)  The  area  of  a  zone  is  30  TT,  and  its  altitude  is  5.  Find  the 
radius  of  the  sphere. 

(6)  On  the  preceding  sphere,  the  sides  of  a  triangle  are  70°,  80°, 
90°.  Find  the  area  of  the  polar  triangle. 

(,-•  875.  If  from  any  point  within  a  regular  tetrahedron  perpendiculars 
are  dropped  to  the  four  faces,  the  sum  of  these  perpendiculars  is  equal 
to  the  altitude  of  the  pyramid. 

876.  Find  the  volume  of  a  regular  tetrahedron  each  of  whose 
edges  is  6  in. 

377.  The  altitude  of  the  frustum   of  a  pyramid  is  7.2  in.,  the 
lower    base   is    10   in.    square,    and   the    upper    base   4  in.  square. 
Find  the  volume  of  the  frustum. 

378.  Find  the  volume  and  the  convex  surface  of  a  cone  of  revolu- 
tion circumscribing  a  regular  hexagonal  pyramid  each  edge  of  whose 
base  is  8  in.,  and  whose  altitude  is  15  in. 

379.  The  base  of  a  right  prism  50  cm.  high  is  a  rhombus,  whose 
longer  diagonal  is  30  cm.,  and  whose  shorter  diagonal  is  16  cm.     Find 
the  total  surface  of  the  prism. 


374  COLLEGE   EXAMINATION   QUESTIONS 

380.  A  leaden  cylinder  of  revolution  10  crn.  long  and  4  cm.  in 
diameter  is  melted  and  cast  into  a  hemisphere.  Find  the  radius  of 
this  sphere. 

881.  The  center  of  each  of  two  spheres  whose  common  radius  is  4 
in.  is  at  the  surface  of  the  other.     Find  the  volume  of  the  solid  com- 
mon to  both  spheres. 

882.  Find  the  locus  of  points  equidistant  from  two  intersecting 
planes,  and  at  the  same  time  equidistant  from  two  points. 

383.  If  two  spherical  triangles  on  the  same  sphere  are  mutually 
equilateral,  their  polar  triangles  are  mutually  equiangular. 

384-  The  bases  of  a  right  prism  5  in.  in  height  are  isosceles  tri- 
angles. The  base  of  each  triangle  is  3  in.  and  the  altitude  is  2  in. 
Through  the  base  of  each  isosceles  triangle  a  plane  is  passed,  inter- 
cepting the  opposite  lateral  edge  of  the  prism  1  in.  from  the  same 
base  of  the  prism.  Find  to  one  place  of  decimals  the  volume  and  the 
total  surface  of  the  solid  cut  from  the  prism  by  these  two  planes. 

385.  If  two  parallel  planes  are  cut  by  a  third  plane,  the  alternate 
dihedral  angles  are  equal. 

886.  The  outside  of  a  glass  ink  well  is  in  the  form  of  the  frustum 
of  a  regular  hexagonal  pyramid,  6  cm.  high,  5  cm.  on  a  side  of  the 
bottom,  and  4  cm.  on  a  side  at  the  top.  The  inside  surface  is  in  the 
form  of  a  right  circular  cylinder,  of  radius  2  cm.,  terminated  by  a 
hemispherical  surface  of  the  same  radius,  the  total  depth  being  4  cm. 
How  many  cubic  centimeters  of  glass  are  there  in  the  well  ? 

387.  A  point  moves  so  as  always  to  be  at  the  constant  distance  x 
from  a  fixed  point  A,  and  at  a  constant  distance  y  from  a  fixed  plane 
MN.     Determine  its  locus,  discussing  the  different  cases  which  may 
occur. 

388.  Prove  that  a  triangular  truncated  prism  is  equivalent  to  the 
sum  of  three  pyramids  whose  common  base  is  the  base  of  the  prism, 
and  whose  vertices  are  the  vertices  of  the  inclined  section. 

389.  Find  the  volume  of  a  truncated  triangular  prism,  if  its  base 
contains  40  sq.  cm.,  the  lateral  edges  are  respectively  5, 10,  20  cm.,  and 
the  projection  of  the  edge  of  length  5  cm.  upon  the  base  equals  4  cm. 

390.  Find  the  area  of  a.  spherical  triangle  if  the  perimeter  of  its 
polar  triangle  equals  297°,  and  the  radius  of  the  sphere  equals  10  cm. 


COLLEGE   EXAMINATION   QUESTIONS  375 

391.  The  radius  of  a  base  of  a  right  circular  cylinder  equals  r,  the 
altitude  of  the  cylinder  equals  h.  Find  the  radius  and  the  volume  of 
a  sphere  whose  surface  is  equivalent  to  the  lateral  surface  of  the 
cylinder.  Construct  the  radius  of  the  sphere  if  r  and  h  are  two 
given  lines. 

,  392.  If  the  solid  angle  formed  at  the  vertex  of  a  triangular 
pyramid  is  a  trirectangular  one,  and  a,  ft,  and  c  denote  respectively 
the  area  of  the  lateral  faces,  and  d  the  area  of  the  base,  prove  that 

a2  +  £2  +  C2  =  ,/2. 

393.  If  a  straight  line  is  parallel  to  a  plane,  any  plane  perpendicular 
to  the  line  is  perpendicular  to  the  plane. 

394>  If  a  plane  is  passed  through  one  of  the  diagonals  of  a  parallelo- 
gram, the  perpendiculars  to  the  plane  from  the  extremities  of  the 
other  diagonal  are  equal. 

395.  A  point  moves  so  that,  if  any  secant  be  drawn  from  it  to  a 
fixed  sphere,  the  product  of  the  whole  secant  and  its  external  segment 
is  constant.     What  is  its  locus? 

396.  If  h  is  the  altitude  of  a  segment  of  one  base  in  a  sphere  of 
radius  r,  the  volume  of  the  segment  is  equal  to  Trk2(r J. 

397.  How  far  from  the  surface  of  a  sphere  of  radius  2  ft.  must  a 
light  be  placed  to  illuminate  one  third  of  its  entire  surface  ? 

398.  Find  the  volume  of  the  portion  of  a  sphere,  lying  outside  of 
an  inscribed  right  circular  cylinder,  whose  altitude  is  h  and  radius  r. 

399.  A  regular  hexagonal  pyramid  whose  base  is  inscribed  in  a 
circle  of  radius  3  in.  has  an  altitude  of  4  in.     Find  the  dimensions  of 
a  similar  pyramid  such  that  its  lateral  area  shall  be  equal  to  the  lateral 
area  of  the  first  increased  by  the  surface  of  a  sphere  of  radius  4  in. 
Answer  need  not  be  simplified. 

400.  A  solid  right  circular  cone  of  altitude  3  in.,  radius  of  base 
2  in.,  rests  base  downwards  in  an  inverted  hollow  cone,  similar  to  the 
solid  cone,  the  axes  of  both  cones  being  vertical  and  the  joint  between 
the  two  water-tight.     If  the  vertex  of  the  solid  cone  just  reaches  the 
level  of  the  rim  of  the  hollow  cone,  find  how  many  cubic  inches  of 
water  can  be  contained  in  the  vessel  so  formed.     Answer  need  not  be 
simplified. 


376  COLLEGE   EXAMINATION   QUESTIONS 

401.  State  and  prove  the  theorems  true  of  the  figure  formed  by 
cutting  through  a  pyramid  by  a  plane  parallel  to  the  base. 

402.  What  is  the  ratio  of  the  volume  of  a  cube  to  the  volume  of  a 
second  cube  whose  edge  is  a  diagonal  of  the  first  cube? 

403.  Find  the  locus  of  the  center  of  a  sphere  which  is  tangent  to 
three  given  planes. 

404.  How  many  rubber  balls  2  in.  in  diameter  and  r\  in.  thick  can 
be  made  from  a  flat  circular  sheet  of  rubber  1  in.  thick  and  2  ft.  in 
diameter  ? 

405.  Find  the  portion  of  the  earth's  surface  bounded  by  a  geodetic 
triangle  whose  sides  are  60°,  90°,  and  100°  respectively. 

406.  Name  and  describe  the  regular  polyhedrons. 

407.  What  portion  of  the  surface  of  the  earth  is  included  between 
the  parallels  of  30°  north  and  south  latitude? 

408.  State  (without  proof)  the  theorem  for  the  area  of  any  spheri- 
cal triangle,  and  illustrate  by  computing  the  area  of  an  equilateral 
spherical  triangle,  each  of  whose  angles  is  70°,  on  a  sphere  of  radius 
5  feet. 

409.  What  kind  of  figure  is  any  section  of  a  cone  made  by  a  plane 
passing  through  the  vertex  and  cutting  the  base  ?  by  a  plane  parallel 
to  the  base  ?     Prove  your  answers. 

410.  If  the  altitude  of  a  cylinder  of  revolution  equals  the  diameter 
d  of  a  given  sphere,  determine  the  radius  of  the  base  of  the  cylinder 
so  that  (a)  the  total  surfaces  shall  be  equal,  (b)  the  volumes  shall  be 
equal. 

411.  Find  the  volume  of  a  spherical  shell  bounded  by  two  concen- 
tric spheres  whose  surfaces  are  20  TT  and  15  TT. 

412.  Enumerate  the  various  cases  of  equal  spherical  triangles,  and 
prove  any  one  of  them. 

413.  In  a  right  triangle  whose  sides  are  a  and  b,  a  line  c  is  drawn 
bisecting  a  and  parallel  to  b.    Find  the  ratio  of  the  surfaces  generated 
by  c  and  the  hypotenuse  of  the  original  triangle  when  the  figure  is 
revolved  about  b. 

414.  Two  truncated  prisms  are  equal  if  three  faces  including  a 
trihedral  angle  of  one  are  respectively  equal  to  three  faces  similarly 
including  a  trihedral  angle  of  the  other. 


COLLEGE   EXAMINATION   QUESTIONS  377 

415.  If  an  equilateral  triangle  and  its  inscribed  circle  are  revolved 
about  an  altitude  of  the  triangle,  then  the  ratio  of  the  volumes  of  the 
cone  and  the  sphere  thus  generated  are  as  9  to  4. 

416.  What  is  the  shortest  line  which  can  be  drawn  on  the  surface 
of  a  sphere  between  two  points  ? 

417.  To  determine  a  point  in  a  given  straight  line  which  shall  be 
equidistant  from  two  given  points  in  space. 

418.  If  a  plane  is  passed  through  a  diagonal  of  a  parallelogram, 
the  perpendiculars  to  it  from  the  extremities  of  the  other  diagonal  are 
equal. 

419.  A  cylinder  of  revolution  is  inscribed  in  a  sphere  of  radius 
6  in.     The  altitude  of  the  cylinder  is  twice  the  radius  of  the  base. 
Find  its  total  surface. 

420.  To  determine  a  point  in  a  given  plane  which  shall  be  equi- 
distant from  three  given  points  in  space. 

421.  A  projectile  has  the  shape  of  a  cylinder  of  revolution  sur- 
mounted by  a  conical  cap.     Its  total  length  is  24  in.,  and  the  cylindri- 
cal part   is  three  times  as  long  as  the  conical  end.     The  greatest 
diameter  is  10  in.     Find  the  volume  and  the  total  surface  of  the 
projectile. 

422.  A  plane  is  passed  through  a  sphere,  bisecting  at  right  angles 
one  of  the  radii.     Compare  the  areas  of  the  two  portions  into  which 
the  spherical  surface  is  divided. 

423.  To  construct  a  sphere  passing  through  two  given  points  in 
space,  and  having  its  center  in  a  given  line. 

424-  Given  a  sphere  10  in.  in  diameter.  A  regular  square  pyramid 
is  inscribed  and  its  altitude  is  8  in.  Find  the  volume  and  total  sur- 
face of  the  pyramid. 

425.  If  the  diameter  of  a  sphere  is  doubled,  in  what  ratios  are  the 
surface  and  the  volume  increased? 

426.  Given  two  intersecting  planes  in  space,  to  determine  the  locus 
of  the  center  of  a  sphere  of  given  radius  which  shall  be  tangent  to 
each  of  the  given  planes. 

427.  One  and  but  one  sphere  can  be  inscribed  in  any  tetrahedron. 


378  COLLEGE   EXAMINATION   QUESTIONS 

428.  In  a  trapezoid  ABCD,  AB  and  CD  are  parallel  and  AC  is 
perpendicular  to  CD.  The  dimensions  are,  AR  =  4  in.,  CD  =  12  in., 
A  C  =  6  in.  The  figure  is  revolved  about  A  C.  Find  the  total  surface 
and  the  volume  of  the  conical  frustum  generated. 

1$9.  The  diameter  of  a  sphere  of  radius  5  in.  is  increased  by  1  in. 
How  much  are  the  surface  and  the  volume  increased? 

430.  A   point   and    any   two   non-intersecting   straight    lines   are 
given  in  space.     Show  how  to  construct  a  straight  line  which  shall 
pass  through  the  given  point  and  intersect  each  of  the  given  lines. 

431.  An  equilateral  triangle  and  its  inscribed  circle  are  revolved 
about  an  altitude  of  the  triangle,  generating  a  cone  of  revolution  and 
its  inscribed  sphere.     Find  the  area  of  the  circle  along  which  the  cone 
and  sphere  are  tangent. 

432.  Find  the  area  of  a  spherical  triangle  whose  angles  are  90°, 
120°,  45°,  on  a  sphere  whose  radius  is  6  in. 

433.  Prove  that  if  the  three  faces  including  a  trihedral  angle  of  a 
prism  are  respectively  equal  to  the  three  faces  including  a  trihedral 
angle  of  another  prism,  and  are  similarly  placed,  the  two  prisms  are 
equal. 

434'  In  a  sphere  whose  radius  is  13  is  inscribed  a  cylinder  whose 
altitude  is  24.  Find  the  ratio  of  the  area  of  the  surface  of  the  cyl- 
inder to  that  of  the  sphere. 

435.  On  a  sphere  whose  radius  is  12,  find  the  altitude  of  a  zone 
whose  area  is  equal  to  the  area  of  the  spherical  triangle  who.se  angles 
are  120°,  110°,  and  80°. 

436.  Prove  that  the  four  diagonals  of  a  parallelepiped  meet  in  a 
point,  which  is  the  midpoint  of  each. 

437.  Prove  that  every  section  of  a  parallelepiped  made  by  a  plane 
intersecting  all  its  lateral  edges  is  a  parallelogram. 

438.  Construct  a  spheric  surface  with  a  radius  r  which  shall  pass 
through  two  given  points  and  be  tangent  to  a  given  plane.     Prove 
your  construction. 

439.  Let  PQ  be  a  line  drawn  from   a  given   point  P,  meeting  a 
given  plane  in  Q.     Find  the  locus  of  the  point  midway  between  P  and 
Q  when  Q  moves  arbitrarily  in  the  given  plane.     Give  proof. 

440-  On  a  sphere  of  radius  10  in.  find  the  area  of  a  zone  with 
altitude  3  in, 


COLLEGE   EXAMINATION   QUESTIONS  379 

441.  A  right  cone  and  a  cylinder  of  revolution  on  the  same  base 
with  radii  3  in.  have  the  same  altitude  4  in.  Find  the  ratio  of  their 
volumes  and  of  their  lateral  areas. 

44%>  Find  the  capacity  of  a  circular  pail  15  in.  high,  the  radius  of 
the  lower  base  being  4  in.  and  the  radius  of  the  upper  base  6  in. 

443.  The  spire  of  a  church  is  a  right  pyramid  on  a  regular  hexagonal 
base,  each  side  of  the  base  is  10  ft.,  and  the  height  is  50  ft.  There  is 
a  hollow  part  which  is  also  a  right  pyramid  on  a  regular  hexagonal 
base,  the  height  of  the  hollow  part  being  45  ft.,  and  each  side  of  the 
base  9  ft.  Find  the  number  of  cubic  feet  of  stone  in  the  spire. 

444-  The  volumes  of  two  similar  polyhedrons  are  64  cu.  ft.  and  216 
cu.  ft.  respectively.     If  the  area  of  the  surface  of  the  first  is  112 
sq.  ft.,  what  is  the  area  of  the  surface  of  the  second? 

445-  Find  the  volume  of  a  cone  of  revolution  the  area  of  whose 
total  surface  is  200  TT  sq.  ft.,  and  whose  altitude  is  16  ft. 

446-  Find  the  area  of  a  spherical  triangle  whose  angles  are  100°, 
120°,  140°,  the  diameter  of  the  sphere  being  16  in. 

44^'  Iu  any  trihedral  angle  the  common  intersection  of  the  three 
planes  bisecting  the  three  dihedral  angles  is  a  straight  line. 

448-  To  what  extent  is  a  straight  line  determined  by  the  condition 
that  it  is  perpendicular  to  (a)  two  intersecting  straight  lines,  (6)  two 
non-intersecting  straight  lines? 

\449-  If  a  section  parallel  to  the  base  of  a  pyramid  is  equal  to  one 
ninth  of  the  base  in  area,  the  altitude  of  the  pyramid  being  6  ft.,  how 
far  from  the  vertex  is  the  plane  of  the  section  ? 

*  450.  A  solid  is  in  the  form  of  a  right  circular  cone,  standing  on 
the  flat  base  of  a  hemisphere  of  equal  base,  radius  10  inches;  the 
volume  of  the  cone  is  equal  to  that  of  the  hemisphere.  Determine 
the*  surface  and  the  volume  of  the  smallest  right  circular  cylinder 
that  will  contain  this  solid,  the  axis  of  the  cylinder  being  the  same  as 
that  of  the  solid.  How  much  waste  space  is  there  in  the  cylinder? 

451.  Explain  why  a  circular  table  with  three  legs,  symmetrically 
placed,  will  be  perfectly  steady  if  placed  on  a  plane  floor,  even  though 
one  of  the  legs  is  slightly  longer  than  the  other  two,  while  a  table 
with  four  legs,  of  which  one  is  a  little  longer  than  the  others,  will  not 
be  steady. 


380  COLLEGE   EXAMINATION   QUESTIONS 


'452.  Two  angles  of  a  spherical  triangle  are  80°  and  120°.  Find 
within  what  limits  the  third  angle  must  lie;  and  prove  that  the 
greatest  possible  area  the  triangle  can  have  is  four  times  the  least 
possible  area,  the  sphere  on  which  it  is  drawn  being  given. 

453.  Define  a  regular  polyhedron,  and  prove  that  there  cannot  be 
more  than  five  regular  polyhedrons.  Assuming  that  there  are  five, 
state  for  each  (a)  the  nature  of  the  faces,  (ft)  the  number  meeting  at 
a  vertex,  and  (c)  the  number  of  faces.  How  many  vertices  has  each 
of  the  regular  polyhedrons  ?  how  many  edges  ?  " 

454-  Prove  that  any  right  circular  cone  can  be  produced  by  the 
revolution  of  a  triangle  about  one  side  as  an  axis.  What  kind  of 
triangle  must  be  used? 

455.  An  irregular  portion  (less  than  half)  of  a  material  sphere  is 
given.     Explain  how  the  radius  can  be  found,   compass   and  ruler 
being  allowed. 

456.  Give  expressions  for  the  surface  and  volume  of  a  sphere  of 
radius  r  and  prove  one  of  them. 

457.  A  solid  sphere  of  metal  of  radius  18  in.  is  recast  into  a  hollow 
sphere.     If  the  cavity  is  spherical,  of  the  same  radius  as  the  original 
sphere,  find  the  thickness  of  the  shell. 

458.  The  volumes  of  a  hemisphere,  right  circular  cone,  and  right 
circular  cylinder,  are  equal.     Their  plane  bases  are  also  equal,  each 
being  a  circle  of  radius  10  in.     Find  the  height  of  each. 

459.  A  sphere  of  radius  5  ft.  and  a  right  circular  cone  on  a  base 
also  of  radius  5  ft.  stand  on  a  plane.     If  the  height  of  the  cone  is 
equal  to  the  diameter  of  the  sphere,  find  the  position  of  the  plane 
that  cuts  the  two  solids  in  equal  circular  sections.     Find  the  area  of 
these  sections.    ~ 

460.  Find  the  area  ^of  a  spherical  triangle  whose  angles  are  100°, 
120°,  75°,  if  the  radius'of  the  sphere  is  7  in. 

461.  The  vertices  of  one  regular  tetrahedron  are  the  centers  of  the 
faces  of  another  regular  tetrahedron.     Find  the  ratio  of  the  volumes. 

462.  Find  the  angle  of  an  equiangular  spherical  hexagon  equiva- 
lent to  the  sum  of  three  equiangular  spherical  hexagons  whose  angles 
are  equal  to  130°. 


COLLEGE   EXAMINATION   QUESTIONS  381 

^463.  The  radii  of  two  spheres  are  13  ft.  and  15  ft.  respectively, 
and  their  line  of  centers  equals  14  ft.  Find  the  volume  of  the  solid 
common  to  both  spheres  (spherical  lens). 

464.  Find  the  altitude  of  a  frustum  of  a  circular  cone,  if  its  volume 
equals  190  cm.,  and  the  radii  of  the  bases  are  respectively  2  cm.  and 
3  cm. 

465.  Assuming  the  earth  to  be  a  sphere  with  a  diameter  equal  to 
8000  mi.,  find  the  area  of   a  zone  bounded  by  the  parallels  of  30° 
north  and  30°  south  latitude. 

466.  The  lateral  surface  of  a  right  circular  cone  is  equal  to  three 
times  the  area  of  the  base.     If  the  radius  of  the  base  equals  r,  find 
the  altitude  and  the  volume  of  the  cone. 

467.  Two  opposite  angles  A  and  C  of  a  spherical   quadrilateral 
A  BCD  are  equal,  and  AB  and  CB  are  produced  through  B  to  meet 
the  opposite  sides  in  E  and  F  respectively.     If  angle  E  equals  angle 
F,  prove  that  AB  =  BC.     Is  the  corresponding  proposition  for  plane 
geometry  correct?     Give  reasons. 

468.  State  a  proposition  that  may  be  used  to  construct  a  plane 
parallel  to  a  given  line.     State  a   proposition  that  may  be  used  to 
construct  a  plane  perpendicular  to  a  given  plane.     Through  a  given 
point  to  construct  a  plane  parallel  to  a  given  line  and  perpendicular 
to  a  given  plane. 

469.  From  the  vertices  of   a   triangle  ABC  perpendiculars  AAf, 
BB',  and  CC1  are  dropped  upon  a  straight  line   XY  in    the   same 
plane.     A  A'  =  2,  BB'  =  3,  CC'  =  1,  A'B'  =  2,  B'C'  =  1,  and  C'A'  =  S. 
Find  the  volume  generated  by  the  revolution  of  triangle  ABC  about 
X  Y  as  an  axis. 

470.  Find  the  volume  of  a  pyramid  whose  base  contains  30  sq. 
cm.  if  one  lateral  edge  is  5  cm.,  and  the  angle  formed  by  the  edge 
and  the  plane  of  the  base  is  45°. 

471.  Prove  that  two  triangles  on  the  same  sphere  that  are  mutu- 
ally equiangular  are  mutually  equilateral.     Are  such  triangles  neces- 
sarily equivalent?     Are  they  necessarily  equal? 

472.  If  a  spherical  quadrilateral  is  inscribed  in  a  small  circle,  prove 
that  the  sum  of  two  opposite  angles  is  equal  to  the  sum  of  the  other 
two  angles. 


382  COLLEGE   EXAMINATION   QUESTIONS 

«/  473.  On  the  base  of  a  right  circular  cone  a  hemisphere  is  con- 
structed so  that  it  lies  outside  the  cone,  and  the  surface  of  the  hemi- 
sphere equals  the  surface  of  the  cone.  If  r  is  the  radius  of  the 
hemisphere,  find  (a)  the  slant  height  of  the  cone,  (b)  the  inclination 
of  the  slant  height  to  the  base,  (c)  the  volume  of  the  entire  solid. 

474-   Find  the  total  surface  and  the  volume  of  a  regular  tetrahe- 
dron whose  edge  equals  8  cm. 

475.  Let  AB  and  CD  be  two  lines  not  in  the  same  plane,  and  let 
E  be  any  point  on  AB  and  F  any  point  on   CD.     What  is  the  locus 
of  the  middle  point  of  the  straight  line  EFt     Prove    the  truth  of 
your  answer. 

476.  The  altitude  of  a  cone  of  revolution  is  12  cm.  and  the  radius 
of  its  base  is  5  cm.     Compute  the  radius  of  the  sector  of  paper  which, 
when  rolled  up,  will  just  cover  the  convex  surface  of  the  cone;  and 
compute  the  size  of  the  central  angle  of  this  sector  in  degrees,  minutes, 
and  seconds. 

477.  Regarding  the  earth  as  a  sphere,  show  that  J£  of  its  volume 
is  included  between  the  planes  of  the  small  circles  of  30°  north  lati- 
tude and  30°  south  latitude. 

478.  If  the  area  of  a  zone  of  one  base  is  n  times  the  area  of  the 
circle  which  forms  its  base,  prove  that  the  altitude  of   the  zone  is 

n  ~     times  the  diameter  of  the  sphere. 
n 

479.  A  variable  sphere  is  tangent  to  a  fixed  plane  at  a  fixed  point 
of  the  plane,  and  a  plane  is  drawn  tangent  to  the  sphere  and  parallel 
to  a  second  fixed  plane.     What  is  the  locus  of  the  point  of  contact? 

J  480.  Prove  that  the  edge  of  a  regular  octahedron  is  approximately 
2.45  times  the  radius  of  the  inscribed  sphere. 

481.  Prove  that  if  the  four  sides  of  a   spheric  quadrilateral  are 
equal,  its  diagonals  bisect  each  other. 

482.  Define  :  (a)    a   perpendicular   to  a   plane ;    (b)  the  distance 
from  a  point  to  a  plane ;   (c)  the  angle  that  a  line   makes  with   a 
plane  ;  (d)  a  right  circular  cylinder ;  (e)  a  spherical  polygon. 

483.  Prove  that  if  a  plane  and  a  line  not  in  that  plane  are  per- 
pendicular to  the  same  plane,  they  are  parallel  to  each  other. 


COLLEGE   EXAMINATION   QUESTIONS  383 

484.  If  from  a  point  P  of  a  perpendicular  PA  to  a  plane  MN  a 
perpendicular  PD  is  drawn  to  a  line  EC  of  the  plane,  any  line  DA 
that  joins  the  foot  D  of  the  second  perpendicular  to  any  point  A  of 
the  first  perpendicular  will  itself  be  perpendicular  to  the  line  BC  of 
the  plane  MN. 

485.  If  two  lines  in  space  are  non-intersecting,  are  their  projections 
on  a  plane  necessarily  non-intersecting  ?    Illustrate  with  a  figure. 

486.  Is  it  true  that  if  two  lines  are  perpendicular  to  each  other, 
any  plane  passed   through  one  of  the  lines  is  perpendicular  to  the 
other?     Give  the  reason  for  your  answer. 

487.  Is  it  possible  for  two  planes  to  be  perpendicular  to  each  other 
if  both  are  perpendicular  to  a  third  plane  ?    Illustrate  your  answer 
by  a  figure. 

488.  Prove   that  if  two  planes  are  parallel  to   a  third  they  are 
parallel  to  each  other.     Is  this  theorem  true  if  the  word  "parallel" 
is  replaced  by  "  perpendicular  "  ? 

489.  Prove  that  if  a   straight  line  is  perpendicular   to  a  plane, 
every  plane  passed  through  that  line  is  perpendicular  to  the  plane.     Is 
it  true  that,  if  a  plane  is  perpendicular  to  a  second  plane,  every  line 
of  the  first  plane  is  perpendicular  to  the  second  plane  ?     Illustrate  by 
a  figure.     Is  it  true  that,  if  a  plane  is  parallel  to   a  second  plane, 
every  line  of  the  first  plane  is  parallel  to  the  second  plane  ?     Give  the 
reasons. 

490.  Prove  that  the  lateral  area  of  a  prism  is  equal  to  the  product 
of  the  perimeter  of  a  right  section  of  the  prism  and  a  lateral  edge. 
Is  this  product  the  same  as  the  product  of  the  perimeter  of  the  base 
and  an  altitude  ?     Give  the  reason  for  your  answer. 

491.  Show  how  to  cut  a  tetrahedron  by  a  plane  so  that  the  section 
shall  be  a  parallelogram.     In  what  three  ways  may  the  cutting  plane 
be  passed  ?     When  is  it  possible  to  get  a  rectangular  section  ? 

\i  492.  The  vertices  of  a  tetrahedron  are  four  vertices  of  a  cube,  no 
two  of  which  lie  on  the  same  edge  of  the  cube.  Prove  that  the 
volume  of  the  tetrahedron  is  one  third  the  volume  of  the  cube. 

493.  Prove  that  if  a  pyramid  is  cut  by  a  plane  parallel  to  the  base, 
the  section  is  a  polygon  similar  to  the  base.  Where  would  such  a 
section  be  placed  if  its  area  were  one  half  that  of  the  base? 


384  COLLEGE   EXAMINATION    QUESTIONS 

494.  If  two  non-coplanar  lines  are  cut  by  three  parallel  planes, 
prove  that  the  corresponding   segments  are  proportional.     If  a   tri- 
angular pyramid  is  cut  by  two  parallel  planes  whose  distances  from 
the  vertex  are  2  ft.  and  5  ft.,  what  can  you  say  about  the  area  of  the 
sections  thus  formed? 

495.  Assuming  that  the  volume  of  a  rectangular  parallelepiped  is 
equal  to  the  product  of  its  base  by  its  altitude,  state  in  logical  order, 
without  proof,  the  theorems  that  lead  to  the  volume  of  any  cylinder. 

49$.  From  a  cylinder  of  revolution  whose  altitude  is  12  in.  and 
whose  diameter  is  10  in.  is  cut  out  a  cone.  The  base  of  the  cone 
coincides  with  one  base  of  the  cylinder,  and  its  vertex  is  at  the  center 
of  the  other  base  of  the  cylinder.  Compute  the  total  surface  and  the 
volume  of  the  solid  that  remains. 

497.  The  area  of  a  certain  small  circle  on  a  sphere  is  equal  to  the 
difference  of  the  areas  of  the  zones  into  which  the  circle  divides  the 
spherical  surface.     Find  the  distance  from  the  center  of  the  sphere  to 
the  plane  of  the  small  circle,  if  the  radius  of  the  circle  is  r.     If  a  cone 
is  tangent  to  the  sphere  along  the  given  circle,  show  that  the  distance 
from  the  center  of  the  sphere  to  the  vertex  of  the  cone  is  ( V5  x  2)r. 

498.  From   a  hemisphere  whose   radius   is   3  in.    a  solid  is   cut 
by  means  of  a  cone  of  revolution  whose  base  coincides  with  the  base 
of  the  hemisphere,  and  whose  altitude  equals  the  radius  of  the  sphere. 
Find  the  volume  and  the  surface  of  the  solid  that  is  left,  correct  to  two 
decimal  places. 

499.  A  hemispherical  dome  140  ft.  in  diameter  is  divided  into  two 
parts  by  a  horizontal  plane  that  lies  seven  eighths  of  the  way  from  its 
base  to  the  summit.     What  will  be  the  expense  of  gilding  the  lower 
part  at  a  cost  of  25  cents  per  square  foot  V 

500.  Show  that  if  a  man  ascended  in  a  balloon  to  a  height  equal 
to  the  earth's  radius,  he  would  see  one  quarter  of  the  earth's  surface. 

501.  Find  the  volume  of  a  spherical  segment  of  one  base  whose 
altitude  is  one  half  the  radius  of  the  sphere. 

502.  Derive  the  formula  for  the  volume  of   a  solid  included  be- 
tween a  spherical  surface  of  radius  r,  and  a  conical  surface  tangent 
to  the  sphere,  the  vertex  of  the  cone  being  distant  2  r  from  the  center 
of  the  sphere, 


COLLEGE   EXAMINATION   QUESTIONS  385 

503.  A  certain  dome  is  in  the  form  of  a  spherical  zone  of  one  base. 
The  base  is  a  circle  24  ft.  in  diameter,  and  the  highest  point  is  9  ft. 
above  the  plane  of  the  base.  Find  the  number  of  square  feet  in  the 
surface  of  the  dome,  correct  to  two  decimal  places.- 

504-  What  proportion  of  the  earth's  surface  lies  between  the 
equator  and  latitude  45°?  Draw  the  figure. 

505.  Find  the  area  of  a  spherical  quadrilateral  whose  angles  are 
135°,  90°,  80°,  and  70°,  on  a  sphere  of  radius  10  ft. 

506.  The  dihedral  angles  of  a  trihedral  angle  are  100°,  65°,  87°. 
The  trihedral  angle  is  closed  by  a  portion  of  a  spherical  surface  whose 
radius   is  4  in.,  and  whose  center  is  at  the  vertex  of  the  trihedral 
angle.     Compute  to  two  decimal  places  the  area  qf  this  portion  of  the 
spherical  surface. 

507.  Prove  that  if  two  arcs  of  great  circles  intersect  on  the  surface 
of  a  hemisphere,  the  sum  of  the  opposite  spherical  triangles  which 
they  form  is  equivalent  to  a  lime  whose  angle  is  the  angle  between 
the  arcs  in  question.     In    what  connection    is   this   proposition  im- 
portant? 

508.  Assuming  the  area  of  a  triangle  as  known,  give,  without  de- 
tailed proof,  the  logical  steps  that  lead  to  the  formula  for  the  lateral 
area  of  a  right  circular  cone.     Does  this  formula  apply  to  an  oblique 
cone?     Give  your  reasons. 

509.  If  a  rectangle  revolves  about  its  shorter  side  «,  and  then  about 
its  longer  side  A,  which  of  the  cylinders  thus  generated  will  have  the 
greater  volume? 

510.  Assuming  the  volume  of  a  cube  as  known,  give,  without  de- 
tailed proofs,  the   logical   steps  which    lead  to   the  formula  for  the 
volume  of  a  right  circular  cylinder.     Does  this  formula  apply  also  to 
an  oblique  circular  cylinder?     Give  your  reasons. 

511.  The  chimney  of  a  factory  has  the  shape  of  a  frustum  of  a 
regular  pyramid.     It  is  180  ft.  high,  and  its  upper  and  lower  bases 
are  squares  whose  sides  are  10  ft.  and  16  ft.  respectively.     The  flue  is 
throughout   a  square  whose  side  .is  7  ft.     How  many  cubic  feet  of 
material  does  the  chimney  contain  ? 

J  512.  The  corner  of  a  cube  is  cut  off  by  a  plane  passed  through  the 
outer  extremities  of  the  three  edges  meeting  at  the  given  corner. 
What  part  of  the  volume  of  the  cube  is  thus  removed? 


386  COLLEGE   EXAMINATION   QUESTIONS 

513.  How  far  from  the  vertex  of  a  pyramid  must  a  section  parallel 
to  the  base  be  passed  in  order  that  the  product  of  this  section  by  the 
altitude  of  the  pyramid  shall  equal  the  volume  of  the  pyramid  ? 

514>    The  volumes  of  two  given  spheres  are  in  the  ratio  1  :  2.     Find 
the  ratio  of  the  total  surfaces  of  their  inscribed  cubes. 
615.    Show  how  to  construct  a  regular  tetrahedron. 

516.  A  right  triangle  whose  legs  are  3  and  4  revolves  about  the 
longer  side.  .  Show  that  the  total  surface  thus  generated  is  less  than 
the  surface  of  a  sphere  whose  diameter  equals  the  hypotenuse  of  the 
given  triangle. 

517.  State  and  prove  the  theorems  true  of  the  figure  formed  by 
cutting  through  a  pyramid  by  a  plane  parallel  to  the  base. 

518.  In  a  sphere  whose  diameter  is  14  in.  the  altitude  of  a  zone  of 
one  base  is  2  in.     Find  the  altitude  of  a  cylinder  of  revolution  whose 
base  is  the  base  of  the  zone,  and  whose  lateral  area  equals  the  area  of 
the  zone. 

519.  Given  two  points  A  and  B  in  two  intersecting  planes  M  and 
N.     To  find  Z  in  the  line  of  intersection  of  M  and  N  such   that 
A  Z  +  XB  shall  be  a  minimum. 

J  520.  What  are  the  ratios  of  the  radii  and  the  volumes  of  two 
spheres,  one  having  81  times  the  surface  of  the  other?  What  are 
the  ratios  of  the  areas  and  radii  of  two  spheres,  one  having  81  times 
the  volume  of  the  other. 

521.  Two  planes  perpendicular  to  the  same  plane  P,  and  containing 
two  lines  AB  and  A'B'  parallel  to  each  other,  are  parallel.     Show 
also  that  this  is  not  true  if  the  lines  AB  and  A'B1  are  perpendicular 
to  plane  P. 

522.  Similar  cylinders  are  to  each  other  as  the  cubes  of  their  alti- 
tudes, or  as  the  cubes  of  the  diameters  of  their  bases. 

523.  Show  that  the  projections  of  parallel  lines  on  the  same  plane 
are  parallel,  but  that  the  converse  is  not  true. 

624-  The  area  of  the  surface  generated  by  the  base  AB  of  the 
isosceles  triangle  OAB,  which  revolves  about  a  fixed  axis  A' Flying 
in  its  plane  and  passing  through  its  vertex  without  cutting  the 
triangle,  is  equal  to  the  circumference  which  has  for  its  radius  the 
altitude  OT  of  the  triangle,  multiplied  by  the  projection  of  the  base 
A  B  on  the  axis  AT. 


COLLEGE   EXAMINATION   QUESTIONS  387 

525.  If,  from  any  point  on  a  sphere  as  a  pole,  with  a  polar  distance 
equal  to  one  third  of  a  quadrant,  \ve  describe  a  circle  on  the  sphere, 
the  radius  of  this  circle  will  be  half  the  radius  of  the  sphere. 

526.  If  M  and  N  are  the  feet  of  the  normals  to  two  planes  from  a 
point  P,  show  that  the  line  of  intersection  of  the  two  planes  is  normal 
to  the  plane  MNP. 

527.  The  base  of  a  prism  of  altitude  7  in.  is  a  regular  hexagon 
whose  side  is  4  in.     The  edges  of  the  prism  make  angles  of  00°  with 
the  altitude.     Find  the  lateral  area  and  the  volume  of  the  prism. 

528.  Through  a  point  A  are  drawn  three  mutually  perpendicular 
lines,  AB,  AC,  and  AD,  B,  C,  and  D  being  any  points  on  the  three 
lines.     The  points  C  and  D  are  joined,  and  from  A  a  perpendicular 
is  dropped  to  this  line  CD,  meeting  it  in  E.     Prove  that  the   line 
joining  E  to  any  point  in  AB  is  perpendicular  to  CD. 

529.  The  three  lines  joining  the  middle  points  of  the  three  pairs 
of  opposite  edges  of  a  triangular  pyramid  meet  in  a  point. 

530.  The  area  of  the  entire  surface  of  a  frustum  of  a  right  circular 
cone  is  120  TT,  and  the  diameters  of  the  parallel  bases  are  8  and  14. 
Find  the  lateral  area  and  the  volume  of  the  frustum. 

531.  The  surfaces  of  a  sphere,  the  circumscribed  right  cylinder, 
and  the  circumscribed  right  cone  whose  axial  section  is  an  equilateral 
triangle  are  in  the  ratio  4:6:9.     The  same  is  true  of  the  volumes. 

532.  In  grading  the  site  for  a  summer  house,  a  mound  is  to  be 
made  in  the  shape  of  a  truncated  right  circular  cone.     The  lower  base 
has  a  diameter  of  12  meters,  the  upper  base  (4  meters  higher),  of  8 
meters.     How  many  cubic  meters  of  earth  will  be  necessary  ? 

533.  If  two  lines  intersect  at  right  angles,  under  what  conditions 
will  their  projections  upon  a  given  plane  be  perpendicular  ? 

534.  Find  the  volume  of  the  frustum  of  an  oblique  pyramid  from 
the  following  data  :  the  lower  base  is  a  square  of  side  4  in. ;  the  upper 
base  a  square  of  side  2  in. ;  one  of  the  inclined  edges,  which  is  8  in. 
long,  has  as  its  projection  upon  the  lower  base  one  of  the  diagonals  of 
that  base. 

535.  Find  the  length  of  the  projection  upon  a  plane  of  a  line  8  in. 
long  (a)  parallel  to  the  plane,  (b)  making  an  angle  of  60°  with  the 
plane,  (c)  perpendicular  to  the  plane. 


COLLEGE   EXAMINATION   QUESTIONS 

536.  Find  the  number  of  cubic  feet  of  concrete  in  a  dam  250  ft. 
long,  31  ft.  high,  33  ft.  wide  at  the  bottom,  and  5  ft.  wide  at  the  top. 

537.  Find  the  volume  of  a  regular  tetrahedron  whose  slant  height 
is  V3. 

538.  Find  the  length  of  wire  ^  in.  in  diameter  that  can  be  made 
from  a  cubic  foot  of  copper. 

539.  The  volume  of  a  regular  square  pyramid  is  43f  cubic  ft. ;  its 
altitude  is  twice  one  side  of  the  base.     Find  (a)  the  lateral  area  of  the 
pyramid,  (6)  the  area  of  a  section  made  by  a  plane  parallel  to  the 
base  and  1  ft.  from  the  base. 

540.  In  a  triangle,  the  sides  including  an  angle  of  120°  are  respec- 
tively a  and  2  a.     Find  the  volume  of  the  solid  generated  by  revolving 
the  triangle  about  the  shorter  side  as  an  axis. 

\  5Jj.l.  A  solid  glass  ball  6  in.  in  diameter  is  expanded  by  a  glass 
blower  till  the  glass  is  1  in.  thick.  Find  the  outer  diameter  of  the 
hollow  globe. 

542.  The  altitude  of  a  cone,  the  diameter  of  its  base,  and  the  edge 
of  a  given  cube  are  equal.     Find  the  ratio  of  the  volume  of  the  cone  to 
the  volume  of  the  cube. 

543.  The  sides  of  a  parallelogram  which  are  12  in.  and  8  in.  re- 
spectively form  an  angle  of  60°.     Find  the  volume  and  the  convex 
surface  of  the  solid  generated  by  the  revolution  of  the  parallelogram 
about  one  of  its  longer  sides  as  an  axis. 

544.  Three  plane  angles  are  20°.  80°,  105°.     Will  these  angles  form 
a  trihedral  angle  ?     Why  ? 

545.  A  cylindrical  water  pipe  has  a  diameter  of  a  inches  and  a  ve- 
locity of  flow  of  b  ft.  per  second.     How  many  cubic  feet  of  water  will 
be  discharged  in   one  minute  at  a  given  point,  no  allowance  being 
made  for  friction  ?     Also,  what  would  be  the  effect  upon  the  quantity 
if  the  diameter  of  the  pipe  were  doubled  ? 

546.  A  circular  sector  whose  angle  is  60°,  and  whose  radius  is  6, 
revolves  about  a  diameter  perpendicular  to  one  of  the  limiting  radii. 
Find  the  volume  of  the  spherical  sector  generated. 

547.  The  volumes  of  two  similar  cones  of  revolution  are  to  each 
other  as  343  is  to  512.     What  is  the  ratio  of  the  lateral  surfaces? 

^  648.  Define  symmetry  as  to  a  center,  and  show  that  the  symmetrical 
figure  of  a  dihedral  angle  is  an  equal  dihedral  angle. 


COLLEGE   EXAMINATION   QUESTIONS  389 

549.  The  total  surface  of  a  regular  tetrahedron  is  60  sq.  m.     Find 
its  volume. 

550.  Find  the  locus  of  those  points  of  a  plane  at  which  a  given 
straight  line,  not  lying  in  that  plane,  subtends  a  right  angle.     Show 
when  the  locus  becomes  a  point  and  disappears. 

551.  The  volume  of  a  sphere  inscribed  in  a  regular  tetrahedron  is 

2  cu.  in.     What  is  the  volume  of  a  circumscribed  sphere? 

552.  Show  how  to  cut  a  given  polyhedral  angle  of  four  faces  so 
that  the  section  shall  be  a  parallelogram. 

553.  What  part  of  the  surface  of  a  sphere  is  illumined  by  a  lamp 
placed  at  a  distance  of  a  diameter  from  the  surface  of  the  sphere  ? 

J  554.  Two  tetrahedrons  which  have  a  trihedral  angle  of  one  equal 
to  a  trihedral  angle  of  the  other  are  to  each  other  as  the  products  of 
the  three  edges  of  the  equal  trihedral  angles. 

555.  A  right  circular  cone  the  altitude  of  which  is  three  times  the 
radius  of  its  base,  and  a  sphere,  the  radius  of  which  is  equal  to  the 
radius  of  the  base  of  the  cone,  are  immersed  in  a  rectangular  cistern 
of  water  the  base  of  which  is  9  ft.  by  11  ft.     If  they  are  removed,  the 
water  level  is  lowered  by  2  ft.     Find  their  dimensions. 

556.  A  cylindrical  tank  10  ft.  long  and  5  ft.  in  diameter  lies  with 
its  axis  horizontal.     If  it  contains  gasoline  to  a  depth  of  15  in.,  how 
many  gallons  are  in  the  tank  ?     (231  cu.  in.  to  the  gallon.) 

557.  Find  the  point  the  great  circle  distances  of  which  from  the 
sides  of  a  spherical  triangle  are  equal.     State  your  construction. 

558.  Four  of  the  six  planes  determined  by  the  diagonals  of   a 
parallelepiped  divide  the  parallelepiped  into  six  quadrangular  pyra- 
mids.    Prove  that  these  six  pyramids  are  equivalent. 

559.  Describe  a  spherical  surface  with  a  given  radius  that  shall 
pass  through  two  given  points  and  be  tangent  to  a  given  plane. 

560.  The  axis  of  a  right  cylinder  passes  through  the  centers  of  all 
sections  parallel  to  the  base. 

561.  The  base  of  a  pyramid  is  a  right  triangle  whose  base  is  12  in. 
and  whose  hypotenuse  is  20  in.     The  altitude  of  the  pyramid  is  15  in. 
Find  the  volume  of  the  frustum  of  this  pyramid  cut  off  by  a  plane 

3  in.  above  the  base. 


COLLEGE   EXAMINATION   QUESTIONS 

562.  The  base  of  a  right  prism  is  a  rhombus,  one  side  of  which  is 
10  in.,  and  the  shorter  diagonal  is  12  in.     The  altitude  is  15  in.     Find 
the  volume. 

563.  A  regular  hexagonal  pyramid  is  cut  into  two  parts  of  equal 
volume  by  a  plane  parallel  to  the  base.     What  is  the  distance  from 
the  vertex  to  this  plane,  in  terms  of  the  altitude? 

564-  Find  the  volume  of  a  right  pyramid  whose  slant  height  is 
13  ft.,  and  whose  base  is  an  equilateral  triangle  inscribed  in  a  circle 
whose  radius  is  10  ft. 

565.  The  volume  of  the  frustum  of  a  square  pyramid  is  74  cu.  in. ; 
the  edges  of   the  bases  are  3  in.  and  4   in.  respectively.     Find  the 
altitude. 

566.  The  altitude  of  a  regular  pyramid  is  2  a,  and  the  base  is  a 
triangle  inscribed  in  a  circle  of  radius  a.     Find  the  lateral  area  of  the 
pyramid. 

567.  Find  the   area  of   a  zone   on  a  sphere  of  radius  r  that  is 
illumined  by  a  lamp  placed  at  a  distance  a  from  the  surface. 

568.  A  cone  of  wood  has  its  vertex  angle  equal  to  60°,  and  the 
radius  of  its  base  equal  to  2  in.     A  cylindrical  hole  of  radius  1  in.  is 
bored  through  the  entire  cone,  the  axes  of  the  two  coinciding.     How 
much  of  the  cone  goes  into  chips  ? 

569.  The  distance  between  two  parallel  planes  is  16  in.     A  line 
24  in.   long  has  an   extremity  in  each  of  these  planes.     Find   the 
length   of  the  segments  into  which  this  line  is  divided  by  a  plane 
parallel  to  the  given  planes  and  4  in.  from  one  of  them. 

570.  What  is  the  greatest  number  of  faces  that  a  convex  polyhedral 
angle  can  have  if  each  face  angle  is  60°?     Why? 

571.  If  the  area  of  a  section  is  one  third  that  of  the  base,  what  is 
the  ratio  of  the  segments  into  which  the  altitude  is  divided  ? 

572.  Define  polar  triangle  of  a  spherical  triangle.     If  a  is  the  side 
of  a  spherical  triangle,  and  A'  the  opposite  angle  of  the  polar  triangle, 
prove  that  A'  +  a  =  180°. 

573.  From  a  point  6  ft.  from  the  surface  of  a  sphere,  one  quarter 
of  its  surface  is  visible.     Find  the  radius  of  the  sphere. 


COLLEGE   EXAMINATION   QUESTIONS  391 

574-  A  triangle  whose  sides  are  respectively  15  in.,  13  in.,  and 
4  in.,  revolves  about  the  shortest  side  as  an  axis.  Find  the  volume  of 
the  solid  generated  by  the  revolving  triangle. 

575.  The  volume   of  a  regular   hexagonal  prism   is  81  \/3;  the 
altitude  of  the  prism  is  equal  to  the  longest  diagonal  of  the  base. 
Find  the  total  area  of  the  prism. 

576.  Find  the  volume  of  a  hemisphere  whose  entire  surface  equals  S. 

±  577.    Find  the  locus  of  a  point  on  a  sphere  that  is  equidistant  from 
two  given  points  on  the  surface. 

578.  The  volume  of  a  sphere  is  4500  TT  cu.  in.     Find  its  surface. 

579.  Find  the  surface  and  the  volume  of  the  solid  generated  by  a 
door  3  ft.  wide  and  8  ft.  high  swinging  in  an  arc  of  144°. 

580.  The  hypotenuse  of  a  right  triangle  is  5  in.,  one  of  its  legs  is 
3  in.     Find  the  volume  of  the  solid  generated  by  revolving  the  triangle 
on  its  hypotenuse  as  an  axis. 

581.  Assuming  that  the  radius  of  the  earth  is  4000  mi.  and  that 
the  crust  is  30  mi.  thick,  find  the  volume  of  the  crust  of  the  earth. 

582.  Prove  that  if  a  line  is  parallel  to  one  plane  and  perpendicular 
to  another,  the  two  planes  are  perpendicular  to  each  other. 

583.  Find  the  weight  of  52,800  linear  feet  of  copper  wire  T\  of  an 
inch  in  diameter.     (1  cu.  ft.  of  copper  weighs  556  Ib.) 

584-  Find  the  number  of  cubic  feet  of  earth  in  a  railway  embank- 
ment 2500  ft.  long,  10  ft.  high,  12  ft.  wide  at  the  top,  and  42  ft.  wide 
at  the  bottom. 

585.  Find  the  cost,  at  $2.50  a  square  foot,  of  gilding  a  hemispheric 
dome  whose  diameter  is  50  ft. 

^,J586.   A  sphere  of  lead  10  in.  in  diameter  is  melted  and  cast  into  a 
cone  10  in.  high.     Find  the  diameter  of  the  base  of  the  cone. 

587.  Find  the  capacity  in  cubic  inches  of  a  berry  box  in  the  form 
of  a  frustum  of  a  pyramid  5  in.  square  at  the  top,  and  4^  in.  square 
at  the  bottom,  and  2f  in.  deep. 

588.  Two  tanks  are  in  form  similar  solids  ;  one  holds  128  gal.,  the 
other  250  gal.     If  the  first  is  20   in.  deep,  find  the  depth  of  the 
second. 

589.  The  total  surface  of  a  cube  is  450  sq.  in.     Find  the  volume. 


392  COLLEGE   EXAMINATION   QUESTIONS 

590.  Define  as  loci  (1)  the  intersection  of  two  planes;    (2)  the 
bisecting  plane  of  a  dihedral  angle ;   (3)  the  plane  which  is  the  per- 
pendicular bisector  of  a  given  sect ;   (4)  the  plane  perpendicular  to 
a  given  line  at  a  given  point. 

591.  Define  (1)  the  angle  between  a  line  and  a  plane;  (2)  a  paral- 
lelepiped ;  (3)  symmetric  polyhedral  angles ;  (4)  the  volume  of  a  geo- 
metric solid. 

592.  Find  the  number  of  cubic  yards  of  dirt  to  be  excavated  in 
digging  a  canal,  50  ft.  wide  at  the  top,  30  ft.  wide  at  the  bottom,  14 
ft.  deep  on  the  average,  between  two  locks  2.6  mi.  apart. 

593.  Compute  the  volume  of  a  regular  tetrahedron  whose  slant 
height  is  \/3. 

594-  The  sides  of  a  parallelogram,  which  are  12  in.  and  8  in.  re- 
spectively, form  an  angle  of  60°.  Find  the  volume  and  the  convex 
surface  of  the  solid  generated  by  the  revolution  of  the  parallelogram 
about  one  of  its  longest  sides  as  an  axis. 

595.  Prove  that  the  smallest  section  of  a  sphere  made  by  a  plane 
passing  through  a  given  point  within  the  sphere  is  that  made  by  a 
plane  perpendicular  to  the  radius  through  the  given  point. 

596.  The  lateral  area  of  a  right  cylinder  is  48  TT  ;  the  volume  is 
96  TT.     Find  the  radius  and  the  height  of  the  cylinder. 

597.  An  isosceles  trapezoid  revolves  about  its  longer  base  as  an 
axis ;  the  bases  are  respectively  14  in.  and  8  in.,  the  legs  each  5  in. 
Find  the  surface  of  the  solid  generated. 

598.  The  radius  of  a  sphere  is  20  in.     Find  the  area  of  a  section 
made  by  a  plane  5.6  in.  from  the  center  of  the  sphere. 

599.  Find  all  possible  locations  of  a  point  that  is  equidistant  from 
two  given  points  in  space  and  at  a  given  distance  from  a  third  point. 

600.  A  cone  5  ft.  high  is  cut  by  a  plane  parallel  to  the  base  and 
2  ft.  from  the  base;  the  volume  of  the  frustum  formed  is  294  cu.  ft. 
Find  (a)  the  volume  of  the  cone,  (6)  the  volume  of  the  part  cut  off 
by  the  plane. 

601.  A  cylindrical  tank  20  ft.  long  and  9  ft.  in  diameter  lies  with 
its  axis  horizontal.  If  it  contains  gasoline  to  a  depth  of  6  ft.,  how 
many  gallons  are  in  the  tank  ?  (231  cu.  in.  to  the  gallon.) 


APPENDIX 

318.  Statements  whose  Conditions  and  Conclusions  may 
have  Two  or  More  Parts.  If  the  condition  or  the  con- 
clusion of  a  statement  is  composed  of  two  or  more  parts, 
the  consideration  of  its  related  statements  can  be  made 
very  complicated  by  taking  those  parts  in  all  their  dif- 
ferent combinations  as  condition  and  conclusion.  The 
full  discussion  of  such  possibilities  has  no  place  in  Geom- 
etry, but  belongs  in  a  course  on  Logic.  The  following 
illustration  will,  however,  show  how  the  different  combi- 
nations can  be  used  to  find  new  propositions.  It  will  be 
considered  only  in  reference  to  the  converses. 

In  an  isosceles  triangle,  a  line  through  the  vertex  parallel 
to  the  base  bisects  the  exterior  angles  at  the  vertex. 

CONDITIONS  : 

The  triangle  is  isosceles, 

The  line  passes  through  the  vertex, 

The  line  is  parallel  to  the  base. 

CONCLUSION  : 

The  line  bisects  the  exterior  angles  at  the  vertex. 
(Since  the  bisector  of  one  angle  also  bisects  the 
other,  this  is  considered  as  one  part.) 

COMPLETE  CONVERSE  :  The  interchange  of  the  entire 
condition  with  the  entire  conclusion  would  give 

If  a  line  bisects  the  exterior  angles  at  the  vertex  of  a  tri- 
angle, (1)  the  triangle  is  isosceles;  (2)  the  line  passes 
through  the  vertex  ;  (3)  the  line  is  parallel  to  the  base. 

393 


394 


APPENDIX 


Of  these  three  conclusions,  (2)  obviously  is  true  ;  (1)  and 
(3)  are  not  true  from  the  condition  alone,  for  the  bisector 
of  the  angle  cannot  affect  the  shape  of  the  triangle,  neither 
is  a  bisector  always  parallel  to  the  base  of  a  triangle,  no 
matter  how  the  triangle  is  drawn.  If,  however,  either 
(1)  or  (2)  is  taken  as  part  of  the  condition,  and  the  other 
is  taken  as  the  conclusion,  a  new  proposition  results,  as, 

In  an  isosceles  triangle,  the  bisector  of  the  exterior  angle 
at  the  vertex  is  parallel  to  the  base.  Or, 

If  the  bisector  of  an  exterior  angle  of  a  triangle  is  parallel 
to  the  base,  the  triangle  is  isosceles. 

In  applying  logic  to  elementary  geometry,  it  is  seldom 
necessary  to  consider  the  general  description  of  the  figure 
as  a  part  of  the  condition.  For  example,  in 

If  two  sides  of  a  triangle  are  equal,  the  opposite  angles  are 
equal.  The  fact  that  a  triangle  is  being  considered  is 
regarded  as  a  general  description,  and  the  statement  is 
therefore  one  having  a  single  condition,  and  its  converse  is 

If  two  angles  of  a  triangle  are  equal,  the  opposite  sides  are 
equal. 

319.  Cavalieri's  Theorem.  —  If  two  solids  contained 
between  the  same  two  parallel  planes  are  such  that  their 
sections  by  any  third  plane  parallel  to  those  two  planes 
are  equivalent,  tlxe  two  solids  have  the  same  volume. 


APPENDIX  395 

i 

Divide  the  common  altitude  of  the  two  solids  into  n 
equal  parts,  and  through  the  points  of  division  pass  planes 
parallel  to  the  base  planes.  Let  M  and  N  be  two  con- 
secutive parallel  planes,  and  let  M  cut  the  solids  in  the 
equivalent  sections  8l  and  82.  At  some  point  in  the 
perimeter  of  8-^  erect  a  line  between  M  and  N  perpendicular 
to  Jf,  and  let  it  generate  a  cylindrical  or  prismatic  surface, 
or  a  combination  of  them,  by  moving  always  parallel  to 
its  original  position,  with  81  as  a  guiding  line.  Do  the 
same  with  S2,  thus  generating  two  surfaces  which  include 
right  solids  between  the  planes  M  and  jy.  These  solids 
have  equivalent  bases  and  the  same  altitude,  and  are 
therefore  equivalent  in  volume. 

If  the  same  operation  is  performed  with  each  consecutive 
pair  of  parallel  planes,  two  sets  of  right  solids  will  be 
formed,  such  that  each  corresponding  pair  are  equivalent 
in  volume,  and  their  sums  are  also  equivalent.  Their  sums 
differ  somewhat  from  the  two  given  solids,  but  if  the 
number  of  divisions  of  the  altitudes  is  increased  without 
limit,  the  sums  of  the  right  figures  will  approach  the 
given  solids  as  limits.  But,  since  the  sums  are  equal 
variables,  their  limits  are  also  equal,  and  the  given  solids 
are  equivalent  in  volume. 

Among  the  various  sets  of  Cavalieri  bodies  are  the  follow- 
ing: prisms,  cylinders,  and  combinations  of  prisms  and  cyl- 
inders, of  equivalent  bases  and  equal  altitudes ;  pyramids, 
cones,  and  combinations  of  pyramids  and  cones,  of  equiva- 
lent bases  and  equal  altitudes ;  a  sphere  of  radius  r,  a  right 
cylinder  of  radius  r  and  altitude  2  r,  hollowed  out  in  conical 
form  from  each  base  to  the  center,  and  a  tetrahedron  with 
two  of  its  edges  in  two  planes  tangent  to  the  sphere  at 
the  ends  of  a  diameter,  and  with  its  midsection  parallel  to 
those  edges  and  equivalent  to  a  great  circle  of  the  sphere. 


396 


APPENDIX 


320.  The  Prismatoid.  The  statement  of  the  formula 
for  the  volume  of  a  prismatoid,  and  the  figures  of  elemen- 
tary geometry  to  which  it  applies,  were  given  in  §  182. 
It  is  now  required  to  prove  the  formula  for  the  prismatoid 
itself. 

Given  the  prismatoid  of  bases  ^  and  52,  midsection  m, 

and  altitude  h  ;  to  prove  F  = -(^ -f  £>2  +  4ra). 

Draw  a  diagonal  in  each  quadrilateral  face  (as  KL), 
thus  dividing  the  entire  lateral  surface  into  triangles. 
Take  P,  any  point 
in  m,  and  by  pass- 
ing planes  through 
P  and  each  lateral 
edge  of  the  figure, 
including  the  diag- 
onals drawn  in  the 
faces,  form  pyra- 
mids from  P  as  a  vertex  to  each  face,  or  triangle  formed 
in  a  face,  as  a  base.  Then  the  prismatoid  is  divided  into 
pyramids  of  which  it  is  the  sum ;  that  is,  P-b1  -f  P-£>2 
+  P-RST  4-  pyramids  from  P  to  each  other  triangular 
lateral  face  =  the  prismatoid. 

On  considering  any  one  of  the  pyramids  from  P  to  a  lat- 
eral face,  as  P-RST,  where  PXY  is  the  part  of  m  cut  off  by 
the  faces  of  the  pyramid : 

P-RST  =  4  (P-XFT),  because  RST  —  4  XYT.    (Why  ?)   But 
P--X"Fr,  with  T  as  the  vertex, 

=  -  PXY,    SO    P-RST  =  ~  PXY. 

6  6 

Since  each  pyramid  from  P  to  a  lateral  triangle  equals 

—-  times  its  part  of  m,  their  sum  equals  -  —  m. 
D  o 


K 


S 


APPENDIX  397 

But        pyramid  P-bl  =  -  bv  and  pyramid  P-62  =  -  52, 

therefore  F  =  -  (^  +  £2  -f  4  w). 

Besides  this  formula  for  the  prismatoid,  there  is  another 
formula  that  uses  but  one  base,  and  instead  of  the  mid- 
section,  uses  the  section  parallel  to  the  bases,  and  two 
thirds  the  distance  from  the  base  used  in  the  formula. 
If  the  section  at  the  two-thirds  point  of  the  altitude  is 

called  s,  this  formula  is  -  (b1  +  3  s).1 

321.  Circular  Cylindrical  and  Conical  Surfaces.  A  cylin- 
drical surface  whose  right  section  is  a  circle  is  a  circular 
cylindrical  surface.  The  surface  of  an  oblique  circular 
cylinder  is  not  circular ;  and  conversely,  an  oblique  cylin- 
der cut  from  the  space  inclosed  by  a  circular  cylindrical 
surface  is  not  a  circular  cylinder.  The  two  ideas,  —  that 
of  the  surface  being  circular,  and  that  of  the  solid  being 
circular,  —  are  therefore  distinct. 

The  formula  for  the  area  of  a  right  circular  cylinder 
can  be  extended  to  any  cylinder  by  making  the  formula 
the  element  times  the  perimeter  of  a  right  section.  It 
will  not,  however,  use  the  circle  in  this  case. 

Similarly,  a  conical  surface  is  circular  if  a  section  per- 
pendicular to  its  axis  of  symmetry  is  a  circle.  The  surface 
of  an  oblique  circular  cone  is  not  circular,  and  an  oblique 
cone  cut  from  the  space  bounded  by  a  circular  surface  is 
not  a  circular  cone. 

The  area  formula  does  not  hold  for  the  general  case  of 
the  cone. 

The  volume  formulas  for  the  circular  cylinder  and  the 
circular  cone  can  be  extended  to  all  cylinders  and  cones. 
1  From  Metrical  Geometry,  by  Dr.  George  Bruce  Halsted. 


398  APPENDIX 

The  formula  for  the  cylinder  is  bh,  and  for  the  cone  is  — , 

o 

where  b  is  the  area  of  the  base  and  h  is  the  altitude.  If 
a  cylinder  or  a  cone  that  is  cut 
from  a  space  whose  surface  is  cir- 
cular, has  not  circular  bases,  its 
bases  are  ellipses,  and  the  area  can 
be  obtained  by  the  formula  7r#5, 
where  a  and  b  are  the  two  half  diameters  (or  axes)  of  the 
ellipse. 

322.  Comparison  between  Plane  Geometry  and  Spherical 
Geometry.  For  many  of  the  plane  geometry  propositions 
concerning  rectilinear  figures  corresponding  propositions 
have  been  proved,  in  spherical  geometry  concerning  great 
circle  arcs.  Corresponding  propositions  do  not  exist  for 
all  such  plane  geometry  propositions  because  two  straight 
lines  must  either  be  parallel  or  intersect  in  one  point,  while 
two  great  circle  circumferences  must  meet  in  two  points. 

This  difference  eliminates  those  theorems  which  depend 
on  parallels,  including  the  propositions  about  similar  fig- 
ures, and  those  which  depend  upon  the  fact  that  two  non- 
parallel  lines  can  intersect  in  but  one  point.  That  it  does 
not  necessarily  eliminate  all  those  which  use  the  single 
intersection  is  shown  by  the  fact  that  two  spherical  tri- 
angles are  congruent  if  they  have  two  angles  and  the  in- 
cluded side  of  one  equal  to  the  corresponding  parts  of  the 
other,  although  in  plane  geometry  the  proof  uses  the  fact 
that  two  lines  meet  in  but  one  point.  In  the  spherical  tri- 
angles the  proof  still  holds  because  the  two  great  arcs  can 
meet  in  but  one  point  on  the  hemispherical  surface,  and  in 
superposing  the  triangles,  but  one  of  the  intersection 
points  can  therefore  be  used. 

One  of  the  groups  of  theorems  most  affected  by  the  two 


APPENDIX  399 

intersections  is  the  one  concerned  with  the  sum  of  the 
angles  of  a  triangle  or  other  polygon.  The  foundation 
proof  of  this  group  is  the  proposition. 

An  exterior  angle  of  a  triangle  is  greater  than  either  in- 
terior angle  not  adjacent  to  it. 

That  this  does  not  hold  in 
spherical  geometry  can  be 
shown  as  follows  : 

In  plane  geometry,  the  line 
AR  is  drawn  from  A  through 
M,  the  midpoint  of  BC,  to  R, 

so  that  MR  =  AM ;  and  BR  is  drawn.  Then  A  MBR  ^ 
and  Z  MBR  —  ^.  C.  But  since  AR  has  met  both  arms  ofZ  CBX, 
it  cannot  meet  either  again,  and  R  and  BR  are  within  that 
angle,  and  so  Z  MBR  is  part  of,  and  is  smaller  than,  Z  CBX, 
which  proves  the  exterior  angle  greater  than  Z  C. 

But,  in  spherical  geometry,  while  all  the  rest  of  the 
proof  holds,  it  is  not  true  that  AM  extended  cannot  meet 
BX  again,  for  it  must  meet  it  at  a  distance  from  A  equal  to 
a  semicircumference.  Therefore 

1.  If  AM  <  a  quadrant,  R 

is  within  Z  CBX,  as  in  posi-  jr      c    \jif 

tion  Rv  and  Z  C  <  Z  CBX.  A£~~~~      "  \* 

2.  If  AM  =  a  quadrant,  R         ^ \*^^>R> 

is  on  AX,  as  in  position  J?2,  B 

and  Z  C  =  Z  CBX. 

3.  If  AM  >  a  quadrant,  R  is  outside  Z  CBX,  as  in  posi- 
tion J?3,  and  Z  C  >  Z  CBX. 

To  show  that  the  exterior  angle  of  a  spherical  triangle  is 
always  less  than  the  sum  of  the  non-adjacent  interior  angles, 
and  therefore  the  sum  of  the  angles  of  a  spherical  triangle  is 
greater  than  a  straight  angle  : 


400  APPENDIX 

If  the  exterior  angle  in  the  last  figure  is  equal  to,  or 
less  than,  the  interior  angle,  no  farther  proof  is  needed  ; 
if  it  is  greater  than  the  one 
interior  angle,  that  is,  if  AM 
is  less  than  a  quadrant,  let 
the  three  great  circumfer- 
ences AC,  AM,  and  AB  meet 
again  at  A1  ',  and  continue  BE 
to  Kon  AC  A', 

Z  CAM  =  Z  MRS  (cor.  pts.   of   ^  A)  and  =  Z  KA'  R  (A 
of  a  lune). 

Z  MRB  =  Z  KRA'     (vertical    A);      .  *  .  Z  KRAf  =  Z  KA*  R, 
and  KA'  =  KR. 

.'  .  KA'  <  KB,  and  Z  KB  A'  <Z.Ar  or  its  equal  Z  A,  and 
Z  ^4  +  Z  U  +  Z  C>  st.  Z. 

This  is  also  shown  by  the  use  of  polar  triangles  in  §  277. 

323.    The  General  Polyhedron.    Theorem.    In  a  polyhe- 
dron of  f  faces,  e  edges,  and  v  vertices, 


If  the  numbers  of  vertices  and  edges  of  any  polyhedron 
are  counted,  beginning  with  any  face,  and  then  taking  a 
second  face,  a  third  face,  and  so  forth,  always  choosing 
a  face  that  has  one  or  more  edges  in  common  with  faces 
already  counted,  and  so  placed  that  the  common  edges,  if 
more  than  one  is  common,  form  one  broken  line,  the  count- 
ing, after  the  first  face,  is  simply  counting  the  sects  and 
vertices,  exclusive  of  the  ends,  of  broken  lines.  This  can 
be  readily  seen  by  experiment  with  a  box,  the  walls  of  a 
room,  or  any  physical  solid  that  is  a  polyhedron, 

In  a  broken  line,  if  the  ends  are  not  counted,  there  is 
one  more  sect  than  vertex.  Therefore  there  are  as  many 
more  edges  than  vertices  in  the  potyhedron  as  there  are 


APPENDIX  401 

broken  lines  to  count.  But  there  is  one  broken  line  for 
every  face  except  the  first  face,  —  where  the  line  is  closed 
and  there  are  the  same  number  of  edges  and  vertices, — 
and  the  last  face,  —  where  there  are  no  edges  or  vertices 
to  count  as  they  have  all  been  counted  in  preceding  faces* 
Therefore,  e  =  v+(/-2),  or  e+2=f+v. 

324.  Radian  Measure.  There  is  another  system  of 
angular  measurement  that  is  sometimes  more  conven- 
ient than  measurement  by  degrees.  In  this  measure, 
the  unit  is  a  radian,  or  angle  whose  subtended  arc,  on  a 
circle  described  with  its  vertex  as  center,  is  equal  in 
length  to  the  radius  of  the  circle.  Show  that  2  TT  radians 
are  equal  to  a  perigon,  and  TT  radians  to  a  straight  angle. 

Since  the  circumference  of  a  circle  has  a  length  equal  to 
2  TT  times  a  radius  (or  6.2832  r),  the  arc  of  a  radian  is 
the  circumference  divided  by  6.2832,  and  the  number  of 

degrees  in  a  radian  is  — ,  or  57.2958°.     In  practical 

6.2832 

use,  this  number  is  seldom  necessary,  for  the  fact  that  a 
straight  angle  equals  TT  radians,  and  also  equals  180°, 
gives  a  more  convenient  comparison  between  the  two 
systems. 

The  use  of  the  radian  in  solid  geometry  will  be  shown 
by  two  examples. 

1.  Using  radian  measure,  find  the  area  of  a  spherical 
triangle  of  angles  90°,  110°,  and  120°,  on  a  sphere  of 
radius  10  in. 

In  radian  measure,  the  angles,  —  compared  with  a 
straight  angle  which  equals  TT  radians,  —  are  J  TT,  -J--J  TT,  f  TT, 
and  their  sum  is  -g6-7r.  But  the  sum  of  the  angles  of  a 
plane  triangle  is  TT  radians,  so  the  spherical  excess  of  this 
triangle  is  -£  ?r  radians.  But  the  area,  A,  is  to  the  area 


402  APPENDIX 

of  the  sphere,  400  TT  sq.  in.,  as  the  spherical  excess  is  to 
720°,  or  4  TT  radians.     Therefore 

x/inA       N         700 

(400  TT)  =  — -  TT  sq.  in. 

2.  If,  on  a  sphere  of  radius  10  in.,  the  sides  of  a  spheri- 
cal triangle  are  12  in.,  10  in.,  and  15  in.,  find  the  area  of 
the  polar  triangle  (to  two  decimal  places). 

The  sides  of  this  triangle  are  1.2,  1,  and  1.5  radii  re- 
spectively ;  therefore  the  angles  of  the  polar  triangle  are 
TT  —  1.2,  TT  —  1,  TT  —  1.5,  radians,  and  its  spherical  excess  is 
the  sum  of  these  angles  minus  TT  radians,  or  (2?r  — 3.7) 
radians.  The  area  is  therefore 

2  ?r -3. 7^^  ^  =  20Q  ^  _  37Q  or  258.32  sq.  in. 

4  7T 

In  the  first  example,  it  is  probably  easier  to  use  degree 
measure,  while  in  the  second,  there  is  little  doubt  that 
radian  measure  presents  less  difficulty.  In  most  cases 
where  the  lengths  of  arcs  are  expressed  in  length  units, 
radian  measure  will  be  found  simpler. 

602.  Find  the  area  of  a  lime  of  angle  —  radians,  on  a  sphere  of 
radius  5. 

603.  Find  the  area  of  Psn  5,,.   7^  ZJT\  on  a  sphere  of  radius  1. 

Vs'  e  '  s  '   3  ) 

604-  On  a  sphere  of  radius  12  in.,  a  spherical  triangle  of  sides 
6  in.,  8  in.,  and  10  in.  subtends  a  trihedral  angle  at  the  center.  Find 
the  face  angles  in  radians,  and  in  degrees. 

605.  In  the  triangle  in  the  last  exercise  find  the  angles  of  its  polar 
triangle,  using  radian  measure. 

606.  How  many  radians  must  the  angle  of  a  lune  equal  if  its  area 
is  half  that  of  the  sphere  V  one  third  that  of  the  sphere  ? 


APPENDIX  403 

325.  Symmetry.  1.  Two  points  are  symmetric  with 
tegard  to  a  center  when  they  are  on  a  line  through  the 
center,  and  are  equidistant  from  it. 

A  figure  is  symmetric  with  regard  to  a  center  when  for 
each  point  of  the  figure  there  is  a  corresponding  point 
symmetric  to  it. 

2.  Two  points  are  symmetric  with  regard  to  a  line  (or 
axis)  when  they  are  on  a  perpendicular  to  that  line  and  are 
equidistant  from  it. 

A  figure  is  symmetric  with  regard  to  a  line  (or  axis) 
when  for  each  point  of  the  figure  there  is  a  corresponding 
point  symmetric  to  it. 

3.  Two  points  are  symmetric  with  regard  to  a  plane  when 
they  are  on  a  perpendicular  to  the  plane,  and  are  equi- 
distant from  the  plane. 

A  figure  is  symmetric  with  regard  to  a  plane  when  for 
each  point  of  the  figure  there  is  a  corresponding  point 
symmetric  to  it. 

4.  Two  figures  are  symmetric  with  regard  to  a  plane, 
line,  or  center,  if  for  each  point  of  one  there  is  a  symmetric 
point  of  the  other. 


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